What settings control the maximum number of connections per user in Oracle 12c? We have some software that is throwing an exception "Pooled connection request timed out"

I know the reason for this and i know the fix. I just need to buy some time on the production server until i can release the software update. 

Currently the software establishes about 640 simultaneous connections before it starts throwing this exception. 

What database settings can i change which will increase the amount of simultaneous connections?
You might want to check the SESSIONS parameter, which controls the maximum number of sessions for the system: 

SELECT name, value 
FROM v$parameter
WHERE name = 'sessions';


https://docs.oracle.com/database/121/REFRN/GUID-52804B5A-164F-44F3-8980-F2593B58D807.htm#REFRN10197

You can also verify that there isn't a limit configured via user profile. Query the DBA_PROFILES view to check if a value is assigned for the SESSIONS_PER_USER parameter associated with your user:

select profile, resource_name, limit
from dba_profiles
where resource_name = 'SESSIONS_PER_USER';


Info regarding profiles:
https://docs.oracle.com/database/121/SQLRF/statements_6012.htm#SQLRF01310