Are you using f:loadBundle? It has problems with ajax since the very beginning.
Try to use a4j:loadBundle instead.
Thanks! It did, in a way, help.
Using <a4j:loadBundle> screws up work of <f:convertDateTime>, which, again, loads its "pattern" attribute from the bundle.
Now I am using both "loadBundle" tags - one - <a4j:loadBudle> - strictly for <rich:calendar>, and the other - <f:loadBudnle> - for everything else.