Python – MySQL – Too many connections?

A script that was working fine on 100 records per batch choked when I fed it 1,000 records.
It gave:

mysql.connector.errors.OperationalError: 1040 (08004): Too many connections

So I went ahead and made sure to connect.close() everywhere that I opened a connection, and the script made it much further but eventually crashed, giving the same error.
Obviously, increasing the connection limit would only encourage sloppy coding? It would be better if I developed a methodology to prevent this type of crash. Because if this were PHP it wouldn't be happening, I admit that.

How can I prevent this type of error from happening again, short of switching to PHP?