Someone interrupting a conversation can mean you feel or fear someone challenging communications in a real-life situation.
Someone interrupting you in the middle of a process, such as reading or having dinner, can mean many things. Consider why they're interrupting you and the context—to tell you some news, they're being disrespectful, they want something from you, they want to interact with you because they like you, etc. Your feelings about the interruption give a big clue about why you were dreaming about it. Your dream may be referring to a recent situation where you were interrupted, or to interruptions from a person or other things in your environment.