No, you can't count anything the Titans did in their previous incarnation, as the Houston Oilers. Nor can you count the Grizzlies in Vancouver, which wouldn't do you any good, anyway.
Honorable Mention to Tennessee college teams: The University of Tennessee Volunteers, the Vanderbilt University Commodores, the University of Memphis Tigers, and, especially due to their contributions to track & field, the Tennessee State University Tigers and Tigerbelles.
Honorable Mention to the Memphis Red Sox. This Negro League team played from 1923 to 1950, and won a Pennant in 1938.
Honorable Mention to the Memphis Mad Dogs. Originally, this was a proposal for an NFL expansion team, put up by William Dunavant, who owned the USFL's Memphis Showboats, and Elvis Presley Enterprises. They wanted to bank on the tenuous Elvis connection (the man had, after all, been dead since 1977), and call them the Memphis Hound Dogs. But when the vote on the 1995 expansion teams came in 1993, it went to Charlotte and Jacksonville.
Before they could decide what to do next, the Presley estate backed out, and the Hound Dogs name went with them. Instead, they became part of the Canadian Football League's brief U.S. experiment, and were renamed the Mad Dogs. They played 1 season at the Liberty Bowl, 1995, went a respectable 8-8, but missed the Playoffs, and were scrapped the next season.
Memphis would finally get the NFL in 1997, when the Houston Oilers became the Tennessee Oilers, playing at the Liberty Bowl. But everyone knew they were moving to Nashville for 1999, and Memphians hate Nashville, partly because it's the State capital and they hated their government, and partly because of the musical rivalry between the cities.
So the fans stayed away, and the Oilers left a year early, figuring they'd still get bigger crowds at the 41,000-seat Vanderbilt Stadium than they were getting in the Liberty Bowl. What's now Nissan Stadium opened the next year, and they've been the Tennessee Titans ever since.
Dishonorable Mention to the Memphis Maniax. They were in the 2001 XFL. They went 5-5, so they weren't a bad team. They were part of a bad idea.
Honorable Mention to the 1970-75 Memphis ABA Team. In 1970, the American Basketball Association moved the former New Orleans Buccaneers up the Mississippi River, and they became the Memphis Pros. In their 1st season, they reached the Playoffs, but got swept by the Indiana Pacers.
After missing the Playoffs in 1972, they were bought by Oakland Athletics owner Charles O. Finley, who redesigned their uniforms from patriotic red, white and blue to A's green and gold, and renamed them the Memphis Tams, for their Tri-State Area: T for Tennessee, A for Arkansas, M for Mississippi. Charlie O knew less about basketball than he did about baseball, going 24-60 and 21-63.
Finley sold the team in 1974, and the new owners made them the Memphis Sounds, redressing them in red and white. They were only 27-57, but that was enough to make the Playoffs, where they lost to the Kentucky Colonels. They folded after the season.
10. 2017 Tennessee Titans. They only went 9-7, but it was enough to make the Playoffs, where they beat the Kansas City Chiefs at Arrowhead, before losing to the New England Patriots at Foxboro. They're only 5-5 this season, so they might not be able to follow this up.
9. 2003-06 Memphis Grizzlies. After failing in their 1st 8 seasons in the NBA -- the 1st 6 in Vancouver, then 2 in Memphis -- the Grizz made the Playoffs 3 straight times, winning 50, 45 and 49 games. But they got swept in the 1st round all 3 times, by the San Antonio Spurs, the Phoenix Suns and the Dallas Mavericks. And they missed the Playoffs the next 4 years.
8. 1984-85 Memphis Showboats. The United States Football League team best known for having University of Tennessee star defensive end Reggie White went 7-11 in their 1st season, but 11-7 in their 2nd, beating the Denver Gold in the Quarterfinal before losing the Semifinal to the Oakland Invaders. We'll never know what they would have done in 1986 or beyond, in either the USFL or the NFL.
7. 2003-08 Nashville Predators. This one is a little tricky, because it's a period that includes the 2004-05 lockout. The 2003-04 season was the Preds' 1st Playoff berth, and they took the Detroit Red Wings to 6 games before losing in the 1st round. The 1st 3 seasons after the lockout, they finished 2nd in the NHL Central Division, but never got out of the 1st round, losing to the San Jose Sharks twice and then to the Red Wings.
6. 2007-08 Tennessee Titans. This didn't last long: A 10-6 season and a Wild Card berth, followed by a 13-3 season and an AFC South Division title, flanked by 8-8 season. They lost their 1st available Playoff game each time, away to the San Diego Chargers and home to the Baltimore Ravens.
5. 2009-12 Nashville Predators. After missing the 2009 Playoffs, the Preds bounced right back with the 3 straight berths. The 2011 season included their 1st Playoff series win, beating the Anaheim Ducks in 6 games, before losing the Western Conference Semifinals to the Vancouver Canucks in 6. The next year, they beat the Red Wings in 5, before losing to the Arizona Coyotes in 5.
4. 1974-75 Memphis Southmen. The World Football League had most of their teams owned by men who didn't know what they were doing. John Bassett was an exception. The Toronto businessman also owned a World Hockey Association team: The Ottawa Nationals, who became the Toronto Toros, and the Birmingham Bulls. He had also been a part-owner of the CFL's Toronto Argonauts.
He had originally wanted his WFL team to be the Toronto Northmen, and raided the 2-time defending Super Bowl winners, the Miami Dolphins, for running backs Larry Csonka and Jim Kiick, and receiver Paul Warfield. But Prime Minister Pierre Trudeau stepped in, and passed the Canadian Football Act through Parliament, prohibiting any U.S.-based league from operating a team in Canada. (It is still in place, although that's not why the Buffalo Bills didn't move to Toronto as threatened. It has been suggested that something could be worked out.)
So Bassett found a big city that loved football but didn't have an NFL team, and they became the Memphis Southmen. The helmet logo was a big brown bear, and local fans, not liking the Southmen nickname, started calling them the Memphis Grizzlies -- which turned out to be convenient for another team that started out in Canada and moved to Memphis, the NBA's Vancouver Grizzlies.
The Southmen went 17-3 in 1974, hanging 60 points on the Honolulu Hawaiians, 49 on the Chicago Fire, 47 on the Jacksonville Sharks, 46 on the Birmingham Americans, 45 on the Houston Texans (for whom the later NFL team would be named), and 37 on the Detroit Wheels. But they also got beat 58-33 by Birmingham, 46-15 by the Philadelphia Bell, and 26-25 by the Portland Storm. They got a bye into the Semifinals, but were beaten by the Orlando-based Florida Blazers, 18-15. The Blazers then beat Birmingham for the only WFL Championship.
The 1975 WFL season was an unmitigated disaster. Even the Southmen didn't escape unscathed: They started 6-1, but lost 3 of their last 4, before the league folded on October 21, 1975 -- a story most people missed because it came on the day of Game 6 of the World Series, the Carlton Fisk Game. The WFL couldn't even die with dignity.
Bassett would be back: He put the Tampa Bay Bandits in the USFL, and they may have been that league's best-run team.
3. 2010-17 Memphis Grizzlies. They made the Playoffs 7 straight seasons. In 2011, they won a Playoff series for the 1st time, beating the Spurs, before falling in 7 games to the Oklahoma City Thunder. In 2012, they lost in the 1st round to the Los Angeles Clippers. In 2013, they had what remains the franchise's best season: They won 56 games, then took down the Clippers in 6 games and the Thunder in 5, to reach their 1st Western Conference Finals. But the Spurs swept them.
They lost to the Thunder in the 1st round in 2014. In 2015, they beat the Portland Trail Blazers in the 1st round, before losing to the Golden State Warriors. They lost to the Spurs in the 1st round in both 2016 and 2017. And then, in 2018, they collapsed, going a terrible 22-60. They may be in the process of shaking that off this season.
2. 2014-18 Nashville Predators. After 2 straight misses, the Preds started a new run, 4 years and currently. They lost in the 1st round in 2015, to the Chicago Blackhawks. In 2016, they beat the Ducks before losing to the Sharks, both times in 7.
In 2017, despite squeaking into the Playoffs as the 8th seed in the West, they almost put it all together: They swept the Blackhawks, their 1st Playoff sweep; they beat the St. Louis Blues in 6; and they won their 1st Conference Championship by beating the Ducks in 6. They gave it a go in the Stanley Cup Finals, splitting the 1st 4 games with the Pittsburgh Penguins, before losing the next 2.
Last season, they won their 1st Central Division title, and beat the Colorado Avalanche in the 1st round, before losing to the Winnipeg Jets. They are currently among the leaders in the Western Conference, but they would have to get to at least a 2nd Finals to jump to Number 1 on this list, ahead of the Tennessee team that, for the moment, has come closer to winning a World Championship than any other:
1. 1999-2003 Tennessee Titans. The 1st 5 seasons in Nashville, 4 Playoff berths, 2 AFC Divisional titles (2000 in the Central, 2002 in the South), and the 1999 AFC Championship, coming oh so close to at least sending Super Bowl XXXIV to overtime. They managed to win postseason games at Indianapolis, Jacksonville and Baltimore, as well as home Playoff games over Buffalo (the Music City Miracle) and Pittsburgh.