〜くて's on the end of verbs at the end of sentences can have a number of different usages which have different nuances, so I think context here is very important. I believe that in speech the style of speaking can also change depending on which usage. The core meaning of it is however "I want to do (something) and (a connotation of something else which is implied)", which can change depending on context.
Was there an ellipsis character (as opposed to linguistic ellipsis as pointed out by @user1205935) after the 会いたくて, as in 会いたくて…? If so, I believe it can change the nuance and I think it could have the meaning of 会いたくて会えない "I want to meet but (unfortunately) can't", but I think 会いたくて。 by itself generally wouldn't have that meaning, and would be used in situations like "I want to meet (so I'm going to)" (unless there's some context I'm missing here).
If there was a "but" before it etc, e.g. でも、会いたくて。 on the other hand I think it wouldn't need an ellipsis character after it to have that meaning.
(I think I also often see this usage of 〜くて with some kind of strengthener before/after it, e.g. 本当に会いたくて…/めっちゃ会いたくて… and can also be doubled for extra effect in songs etc, e.g. 会いたくて、会いたくて…).
Edit: Tried to make things a bit clearer.