The impression that the Cardassians weren't a major power comes from how easily the Klingons beat them in DS9 season 4. One assumes the Federation would be able to do this too. That they had to join the Dominion to be taken seriously militarily also lends weight to this. The Border War that look place before TNG grays things a bit but I like to think the Federation just didn't put all their effort into it and settled for a doctrine of containment rather than defeating them. Though their losses during the Dominion War clearly leave them as a second rate power and a dependency of the Federation after the war.
I was always under the impression that the Klingon/Federation alliance presented a super power that no one empire in the quadrant would be able to contend with. The impression that the Klingons are weaker than the Federation can be had easy enough from DS9's seasons 4 and 5 where the Federation was pretty much able to check them. Couple that with their losses during the Civil War that took place in TNG. Sloan also mentions that the rate of Klingon casualties during the Dominion War would leave them a second rate power.
The Romulans by there very nature were a vague thing. They'd skirmish regularly with both the Klingons and the Federation but wouldn't push it to all out war. So that would imply that they're either on equal footing or somewhat stronger than the Federation, but not powerful enough to confront both them and the Klingons. Sloan stated in the series that the struggle for local dominance after the Dominion War would be between the Federation and Romulans, which I think is credible. They probably look lower casualties during the war too being it wasn't fought in their space and they didn't join it until later. The only hint of weakness comes after it. One assumes they took a lot of military casualties during the Reman Insurrection and the whole Hobus Supernova is bound to ruin their long term power projection ability.
I really never got the impression that the Federation was pulling way ahead of the pack until the late movie era and novelizations that follow them.