Shared hosts put a bunch of accounts on the same server. As such, you're site is subject to the "bad neighbor effect". What this means is your site might slow down if the server is overcrowded or overstressed by other sites.

Good hosts won't stress their machines too much, but it can happen at any shared host. The only way around this is to buy hosting that guarantees you resources on the server. This can be done with a VPS/VDS plan (virtual private server), or with a true dedicated server.

Dedicated servers generally start at $100/month, but you can get decent VPS service for around $20/month. Please note that a dedicated server, or VPS requires you to administer the server yourself. This often means configuring apache, PHP, etc on your own.

I really like 6sync's VPS, and am using their miniVPS for many of my sites. Great, affordable VPS and comes with full management for those that are a little unsure about a VPS, but you still need to know some server admin skills, or be willing to learn them.

If you don't need any support, and just want the most bang-for-your-buck, I'd recommend ServerAxis. I've been using them for a couple months and am pleased so far.