Are you able to connect to the SQL Server instance locally?
The error you get is typically the one you get when you have run out of worker threads, but in that case, you should not able to connect locally either, I think.
If you can connect locally, what does
sp_configure 'user connections'
report for the third and fourth columns? Anything but 0 is suspect!
What about this one:
SELECT * FROM sys.dm_os_wait_stats WHERE wait_type = 'THREADPOOL'
What are the values on both instances?