two things could cause the errors you mentioned. One has already been identified, increased traffic requiring an upgrade to a VPS. While MANY things are Unlimited on sharing, memory and CPU usage are not. When those are getting used up, the next step up is a VPS. Watch you VPS usage closely tho and/or install ottodv's PS Manager, or you could end up paying as much or more as you would for a dedicated server.
Secondly and unmentioned so far, if you have made recent changes to plugins or themes, try undoing those changes. Plugins in general can get memory hungry alone, and then to make it more confusing and harder to figure out, certain COMBINATIONS of plugins will cause things to thrash back and forth and eat up more memory and CPU than the idvidual plugins would use when used alone.
In other words, you mught be using plugin A and B and never have a probolem, but add plugin C and then plugin B will no longer function efficiently. Alot of this is trial and error, but if you post the plugins you are using im sure you will get some feedback.
Also, if your not using a plugin called login lockdown, get it and install it. It blocks hackers from attempting to crack your admin login. The hackers can eat your memory and CPU trying...even if they never get in.