Is it possible to unlock every airport in Pocket Planes?

Yes. Going by the level up bonus available in the most current update of the game, you receive 2 new airports every time you level up. The game starts on level 1 with you having the ability to build 5 airports.

This means that the number of airports = 5 + 2 * (level - 1). There are 251 airports that you can unlock. In order to do that, you will need to be level 124.

However, the number of XP required to level up increases at a non-linear rate, so reaching level 124 will take quite a bit of gameplay.

As an aside, you will find that, unless you have a huge number of airplanes, it doesn't always make sense to keep a lot of the smaller airports open. (See here for example.) So just because you can have all of the airports unlocked, doesn't mean it's a good strategic idea to do so.