I am on FreeBSD, I am on a rent server, running dozens sites, and always using mysql_connect. Just request the hosting company increase the connections limits.
I don't have page takes a long time to run. Haven't see the problem for one year. No new big sites added to the server, no sudden increase in traffic.
I had this php connections problem one year ago before, after the hosting company increase memory and increase the connections limits, I hadn't seen this problem for a year.
But recently, I had seen this exceeding the limits connnections problem 2 times, the hosting company is going to update the system, but at the same time, they suggested me to change the connect to pconnect. I think it was wrong that change connect to pconnect, (the hosting company may have the old knowledge like php before 4 which doesn't close itself).
Also, I have seen other problems, such as DNS service problems, or server running out of the air shut down eventually. I think it was not mysql limit connections, it was the server problem. But I just want to investigate the mysql_connect and mysql_close() etc. just to be on better programming practice if I need to be.