Messaging Limits
Page Rate limits are in place to prevent malicious behavior and poor user experiences.
Page Rate Limit (Per day ) = 200 * Total Messenger Audience Or maximum of 250 messages/sec
Note:Total Messenger Audience is the number of people your business (Page) can send messages to. If you have multiple pages connected, since the rate limit is based on the Page audience, it is possible for it to hit the rate limit for one Page and not another.