Database connections are a limited resource. Some DBs have a very low connection limit, and wasting connections is a major problem. By consuming many connections, you may be blocking others for using the database.
Additionally, throwing a ton of extra connections at the DB doesn't help anything unless there are resources on the DB server sitting idle. If you've got 8 cores and only one is being used to satisfy a query, then sure, making another connection might help. More likely, though, you are already using all the available cores. You're also likely hitting the same harddrive for every DB request, and adding additional lock contention.
If your DB has anything resembling high utilization, adding extra connections won't help. That'd be like spawning extra threads in an application with the blind hope that the extra concurrency will make processing faster. It might in some certain circumstances, but in other cases it'll just slow you down as you thrash the hard drive, waste time task-switching, and introduce synchronization overhead.