This sounds like the type of message one gets when they send messages to large groups of people not just one person. Well unless they were rapid fire and the other person was not responding but in a one on one conversation...?
What app or service are you using to send these messages?
When you say short, what do you mean? One or two words, one or two lines?
Could you estimate for us how many messages comprise the messages sent at that time?
Is this a group conversation all in one thread? If yes, are they set to go out and come back as one conversation or do you have it set to send individual messages to each so that everyone doesn't get responses from everyone else?