Thereâ€™s two ways to fix the problem. First, if you find that you only hit the limit during mysqldumps and never during normal database operation, just add --single-transaction to your mysqldump command line options. This will cause mysql to keep only one table open at a time.
However, if this happens while backups arenâ€™t running, you may want to increase the open_files_limit in your MySQL configuration file. By default, the variable is set to 1,024 open files.
For further reading:
5.2.3. System Variables
7.13. mysqldump – A Database Backup Program