here is a really good example:
That dude would knock out all the MTF readers for sure. A gut doesn't hurt your strength one bit. Just means you have eaten more energy than you spent. It does have some advantage in physical sports as it "shields your organs and bones" in some sense. It acts as a cushion to damage. But as noted above it definitely hurts your endurance.
Why: because you are heavier and have a bigger body volume = more strain on the heart. Heart has to work harder and you get tired faster. Also more strain on your knees and joints = higher risk for injury.
So it depends on what the critera for fit is. Is it just being atheletic then sure you can have a gut. Is it being very agile with great endurance (important for soccer, tennis, long distance running etc etc) then having a gut is definitely sign of major fail and cheating with the training.
For most normal people it's more a question about diet than actual training. Fat doesn't disappear with training really. It's the body reserve. Build more muscle and your body consumes more energy so unless you start to eat more then your gut slowly disappears. Stop training and your muscles will wither away and unless you start to eat less to compensate for lower energy consumption then the gut comes back to say hello
I don't think Roy Nelson is a proper example for this thread, sure the guy has insane power + amazing chin + actual boxing/BJJ skills. Now, is he a FIT guy? Hell no. He looked pathetic in the coaches' challenge for the Ultimate Fighter were he was the team leader and he's well known for gassing awfully after the first couple of rounds, sometimes after the first round. He's clearly not a good example, and definitely not successful due to his fitness or stamina.
Best example that came to mind is Cain Velasquez:
Sure, he doesn't have a huge prominent gut but, not only he still has one, he's also got a very high bodyfat % and he's not ripped/shredded at all. Nevertheless, he's got remarkable cardio for a heavyweight (approx 110kgs) and can keep an insane pace during long fights with nonstop action.
On topic, yes it is possible to be fit while having a gut but it's far from a rule and some people are just naturally amazing athletes (exploited after the proper training of course).