Doctors do always tell you to finish your course of antibiotics, even if you feel better. The reason for this is that you feel better before all the bacteria are killed within you body. If you stop the treatment early, the bacteria will start to grow again and the infection will recur. It is also likely that the second attack will consist of bacteria with more immunity to the antibiotic (the ones without any resistance were killed, right?) and so will be worse. However there are other reasons apart from the immunity issue. Here is a good article on the subject. And another. And another. And another.
Other factors also contribute to the rise of antibiotic-immune bacteria, including the overuse of antibiotics for minor illnesses.
You edited to ask if bacteria "become immune" to the antibiotics. Its not clear what you mean by "become immune" but the answer is still probably "yes". The second wave attack by resistant bacteria is due to two effect. Selection is the primary one. Some of the bacteria in the infection are more resistant to your prescribed antibiotic than others. A partial course removes the bacteria that have low resistance and leaves those with high resistance, which form the 'second wave' infection that results from stopping the course early. In that case those bacteria are not "becoming resistant", it's just that the resistant ones predominate. However it is also true that mutations can and do occur, and this can result in bacteria that previously had low resistance becoming highly resistant. In that sense bacteria sometimes are "becoming resistant". This paper gives more information.