I just had this issue today, I have the same environment as the one you have described.

Seems like Analysis Engine can't figure out how many parallel Processes it can run to handle the Cube processing, seems like a bug to me !

ole db error: ole db or odbc error: the query processor could not start the necessary thread resources for parallel query execution.; 42000.

So to work around it, I went under "Change settings" and forced the parallel execution to 4 and it is now processing... i hope it will go back to normal after help desk installs the pending patches or it seems that I will need to restart SSAS services