If there's a line a people waiting to be served, to cut in is used if you go ahead of someone that was there before you. The term, "Cutting in line" refers to this. It's never a good idea to cut in line and fights can start if someone does it.
Cut in can also be used if there's a conversation and someone rudely starts talking over another person. They're cutting the other person off and have therefore "cut into" the conversation.
To break in usually means personal property has been disturbed. A thief breaks into places to steal things. For example, "Someone broke into my car and stole my radio!" "My house had an attempted break in, but the alarm system went off and the burgular ran away."