If the destination airport is very busy under normal weather and then poor weather requires a diversion to your flight planned alternate then you can expect many other flights doing the same. This often results in your alternate airport filling up very quickly with other diverted aircraft resulting in that airport not being able to accept any more diverts because either there is no parking space available or the ground handling companies or Terminal capacity can't cope.
When selecting an alternate make sure your airline has a contract agreement in place with a ground handling company at that airport otherwise you could wait a very long time
for handling. Even if you you did have a contract handling arrangement in place it can still take a long time to be handled as other non-diverted scheduled flights at the airport may take priority.
There are many variables that result in alternate airports not accepting diversions.