I would guess that, for the OP’s thread purpose, it is just probability-wise much more common for someone to be a fan of a team that’s never won a championship in the time they were around to witness it, than not. Each sports fan probably supports somewhere between 3-7 different teams. More likely than not, at least one of those teams will be in the midst of a multi-decade championship drought.
As they say: bad timing. 
I’m guessing that you’re either too young to remember the Cowboys’ championships, or you just didn’t care about them back then.
Either way, those of us who are of a certain age, who support other teams, probably feel that the Cowboys’ relatively-recent drought would be just fine if it went on for another decade or three. 
I was born in 1962, but didn’t move to Canada until 1968, the year after the Leafs last Stanley Cup. I’m still waiting, but they have built a very good team and I would be shocked if they don’t win the cup in the next 2 to 5 years.
Thankfully the Blue Jays won back-to-back World Series in '92 and '93.
Those are the only two teams I’m passionate about.
I was too young and didn’t get news about the Cowboys before 1999 - in fact I didn’t even know the Cowboys won Super Bowl XXX in 1996 until several years after it happened.
Not since the Cubs won the World Series. I’m a fan of the Cubs, Chicago Bears, LA Lakers, and Florida Gators. You could make the case that the Gators haven’t won championships in every sport that they sponsor, but they have won national championships in men’s basketball and football (along with baseball in 2017!)
I pretty much grew up around Kankakee and graduated high school in New Orleans.
The Detroit Lions. I’d be happy with a single post-season win. The last time they won a playoff game was 1991. Super Bowl victories is just crazy talk.
That’s The Curse Of Bob Short. He stole the Senators away from Washington in 1971. They will never win a championship until at least after the Nats do.
The **Knicks **are my Basketball team, last win was from before I followed anything but baseball, 1973.
**Rangers **were my Hockey team and they did win once in 1994 but I dropped hockey about 20 years ago.
Much better with Football and Baseball, Giants & Yankees respectively.
No team I support has ever won a championship while I have supported them.
I adopted the Nationals as my favorite team in 2005. I lived in NY from 1993-98 and rooted rooted for the Yankees, so I got to see my team win a championship there. I don’t follow hockey or football as closely, but I’ve gotten to see the Ravens win twice and the Caps once.
The Detroit Lions last won a championship in 1957.
I was born in 1958.
Some call it the Mustard Curse.
mmm
Not I. My only rooting interest has been the Kansas City Royals, who won both of their championships since I became a fan (in 1982).
Though I did follow them through some very very bad stretches of play.
Since the Eagles won every professional team I support has won, though I’ve watched the Flyers lose 6 straight Stanley Cup finals since the last time I saw them win one.
For college I’ve watched my school lose in the Lax championship game twice, each by one heartbreaking goal, and saw the hockey team lose in the NCAA Semifinal. No championships there.
Haven’t Won:
Kansas City Chiefs
St. Louis Blues
Arsenal (started from their CL final in 2006, they have won few FA cups since then)
Have Won:
St. Louis Cardinals (should I point out to those Rangers fans above that I was at 2011 WS Game 6 in St. Louis, the undisputed pinnacle of my sports-watching life?)
St. Louis Rams (only a quasi-fan)
Another Browns fan, here. 'Nuff said.
I’ve been fortunate to have seen all of the teams I’ve followed win a championship, but there is a caveat at the end…background, I’m soon to be 56, born and raised in Baltimore, and while I’ve lived in upstate NY for the last 19 years, we remain serious Baltimore and Maryland fans, and our grown boys (who spent their formidable years in NY) are also the same…
Baltimore Colts (1970)
Baltimore Ravens (2001 and 2013) For the record, I was a junior in college when the Colts left town, I grew up hating the Redskins, hated the fact the Colts left, so was never a Indy fan…ever…
Baltimore Orioles (1966, 1970 and 1983) 1966 I was four, I don’t really remember them winning, I had no concept, but I do remember my father and a bunch of neighbors screaming and dancing in the street…
Maryland Terrapins BB (2002) This one was very bittersweet, my Dad was a lifelong Terp fan, passed it down to me, he went MD law school, I went to grad school there, he died in February 2002, they won about 6 weeks later…
Love Terp lacrosse, grew up playing, watching MD and Hopkins play live in the '70’s as a kid was an awesome experience…their most recent Championship (2017) was fantastic…
So, the caveat, my boys have celebrated a couple of Ravens SB wins, the last one was great as they were 17, 19 and 24 at the time…they are also rabid Oriole fans, and I’m pretty sure the Orioles will not win another WS in my lifetime, and that’s something I’d really love to celebrate with them…yikes…they are really bad and it’s not looking good anytime soon…
St. Louis Blues
Kansas City Chiefs
Royals would be on the list if not for their World Series win a few years ago. (I was still more of Cards fan than Royals in 85.)
Honorable mention goes to the University of Kansas football team. Talk about perennial losers…
Atlanta United may be Atlanta fans’ best hope.
As to the original question, the Falcons, Atlanta United (but they’ve only been in existence for 2 years), USMNT, Newcastle United, The New York Mets (I’m from Jersey, and was 6 when the Mets won their last World Series).
The Indians and the Browns, teams that I’ve followed since I was old enough to read the sports page, have won nothing. But the Carolina Hurricanes and the Seattle Sounders, two teams I’ve picked up following as an adult, have both won championships. I’m not sure what that means, but I’m positive it’s something deep and telling about my soul and the nature of growing up. Or something.