This is something that truly depends upon the tone of voice and facial expression when spoken. Most of the time, when someone is described as "out of his/her league" it means that they'll never measure up to that person - he/she is too good-looking, too athletic, too smart, etc. For example, I might see a George Clooney look-alike at a party, and my friend might encourage me to go talk to him, saying "He's here alone, he didn't bring a date, go say 'hello'". And I would smile ruefully and say in a defeated tone of voice, "No way, I'm totally out of his league" meaning that I'm no supermodel and this man would never give me a second glance.
However, at that same party, if I approach a person whom I know is single and say "You should come meet Eugene, he's really nice and he's here alone" and that person looks over at Eugene, decides that he's a nerd and says "I am out of his league" in a haughty tone, accompanied by a look of minor revulsion, it indicates that this person thinks she is better looking/smarter/richer (etc) than Eugene and to befriend him would be beneath her.
Student or Learner