Alliances
From EU3 Wiki
The alliance system in EU3 is bilateral, which means that all alliances are treaties between two countries. This means that it is perfectly possible for a country A to be allied to countries B and C that are bitter enemies. Country A might also come in a situation where they have to choose which side to support in a war.
If you seek an alliance with another country and it says "impossible", there are several reasons this could be the case.
1) They already have 3 allies. AI players are limited to 3 alliances, the player can have an unlimited number
2) Your Relations with the other nation are -100 or worse
3) Your Reputation affects your chances to make diplomatic moves
4) Your Prestige score affects your ability to attract allies
5) The diplomacy rating of your ruler
6) If trying to ally with a Vassal, there must be a route from your capital to theirs. The route can cross the sea, but not other countries (even your other vassals) and detached provinces don't count - it has to be capital to capital. For overseas vassals, you must have a connection to a coastal province so you can connect with overseas vassals.

