The default max_connections limit is 151 which should be enough for most servers. If you're hitting it there is usually an underlying cause - either the server is CPU or disk i/o limited or maybe the disk is full. Increasing the limit might exacerbate the problem.
Another possibility is that a site is holding on to connections for too long - it's opening a connection then doing something slow like connecting to an external API whilst holding on to that connection.
Instead you might want to look at setting max_user_connections.