There is nothing within the database or code that would prevent a message count from increasing. Technically XF supports a message_count value that exceeds 4.2 billion (4294967295 to be exact).
So the usual troubleshooting steps would have to apply. Do you have an add-on which may cause the message count not to update in some cases? Have you enabled a setting in any particular forum that does not increase the message count in that forum? If other users post in the same forum does their post count increase? If you disable all add-ons and use the default style does the message count then increase when they make a new post?