This is a romantic comedy on the surface, and it's not a bad one at all, with sharp dialogue, surprising transitions where the characters switch from being the cat to being the mouse and vice-versa, and dancing and music and fun and silliness.
I also found it surprisingly poignant. It covers a lot of the same ground as films from the same period like "The Clock" and "Since You Went Away" - a compressed courtship between a soldier and a civilian, where they have a very short time between meeting as strangers and the soldier going off to war. These films (which aren't just Hollywood fantasies, they would have been happening to millions of people in real life) have two sources of dramatic uncertainty - firstly the uncertainty about whether they're really getting to know each other or they're just on an emotional roller coaster; and secondly the uncertainty about whether it's fair to get married and run the risk of the civilian being left a widow or spending the rest of her life looking after a severely injured husband. These issues aren't explicitly discussed in "The Sky's the Limit", which is still a romantic comedy, but they're alluded to sufficiently clearly that a contemporary audience would have understood that Astaire's character was very confused, unsure about whether to hit the accelerator or the brake, and wound up enough that he could have gotten drunk and smashed up a bar.
Another striking scene in the movie was a comment Astaire's character made about how one might go to war not for any grand cause but to preserve one's freedom to be a slacker. He was behaving consistently with that declaration in (at least initially) wanting to spend a few days out of uniform, joking around and having fun with a pretty girl. There are questions about whether an actual WW2 fighter pilot on leave would behave that way - I don't know, within the film, I find it plausible enough for suspension of disbelief, and if nothing else it's a nice way of inserting a "why we fight" message about the United States not being a nation of full-time uniformed soldiers, but of civilians who occasionally put on a uniform to defend life, liberty, and the pursuit of happiness.
