Dreamhost problem? Maybe, maybe not.
Sites do require maintenance, and that is the customers job. Like a car, you have to regularly check the gas and add to it, less regularly you have to check other things and maybe add some of this or that, or replace a piece or two, or upgrade the tires...etc.
Well you say, "but we haven't done anything to our site, this just happened", well that may be true, but the software that runs the server has evolved. Likely there's been updated to the software you are using as well. or maybe something even more evil, like hackers are attempting to find exploits and your domain has become a target. (if you watch the http and error log you will be amazed how often these attempts occur.)
or maybe it could be an issue with a neighbor on the shared host or shared IP or some other dreamhost problem.
Your question is essentially is this a dreamhost thing? but with the info you provided there is no way to tell. As far as comparing to others on shared hosting to see if its a common issue, that would be very doubtful, there are 100's of shared servers at dreamhost and each of those has many many sites.
Can you improve your site by upgrading hosting? Sure, most likely..... but like I started out, there is no free lunch... whatever hosting your on, you'll have to dig in and figure out what's up when a problem pops up, and that will happen no matter what hosting you have.