Raw logs for your domain are available via ssh/SFTP at /home/USER/logs/DOMAINNAME/http/ (hint: you can NOT navigate there via FTP, you MUST use a secure protocol or the path won't exist or will be empty.)
As far as a problem with the server itself, from what you said probably not. Of course one of the things worth checking immediately is the servers load averages using
top. Likely you won't find a problem there, dreamhost must pay attention because you don't see many complaints in this forum about overloaded shared servers. If you do find high load averages, you likely can't do anything to fix it other than open a ticket.
So let's move on the the next layer, the webserver (or apache instance). This is likely where the issue is/was. Apache instances are also shared. In other words, each shared server has multiple shared apache instances. This is also the level at which IPv4 IP's are assigned, that is each website that shares an apache instance also shares an IPv4 address. A DDoS attack against another site could be felt here, but that shouldn't last too long tho because dreamhost defends against them routinely and has protocols in place to help protect other customers.
There was a long running "apache glitch", which I hope using the word "was" is correct--we certainly hope that didn't migrate from debian to ubuntu--no word from dreamhost about the status of that bug. Again tho, if the problem is at this level you probably can't fix it, except to open a ticket. You can only see the http logs for your domains, if someone elses site on the same apache instance is under attack you can only guess.
You can use tools, like this one, to find guesses as to what sites are sharing the same IP, beware tho the way data is collected tho makes it incomplete and/or out-of-date by definition--manually verify the IP of any site you might be sharing with, it may have moved or no longer exist. That said, I did use information gathered via this method once in a support request to move my site to a different apache instance. I suspected by name that one site might be susceptible to attacks on its subject matter. Changing the apache instance in that case did solve the problem, in retrospect tho it might have just been that apache glitch we already talked about. Along the way I learned that if you request that your site be moved to another apache instance to eliminate that from a list of possibilities that support is very cooperative and even will hand pick the new apache instance.
Did you open a ticket? Of course if the problems not happening anymore support can't very well chase it. If it's an intermittent bug you have to try to find the pattern.
If you have cloudflare turned on, that changes everything--most users requests don't reach the server directly.
[EDIT: my memory failed... I don't think this was you, I can't find the thread I was think of tho... I'll leave it since it's good info for someone reading along anyway.]
Also, since you mentioned elsewhere that your wife wants you to set up a blog for her, certain combinations of wordpress plugins and/or theme's can cause issues like you described in your post. Not everything plays nice together. If you need plugin X for reason Y, that may preclude you from also being able to use plugin Z. If your dealing with diagnosing a wordpress domain, then first thing to do is disable all plugins and change the theme back to twenty-fourteen. If the problem goes away turn things back on one at a time until you find where conflicts start, and/or just ask around. Most of the time if you ask in this forum or over in the wordpress forum for specific plugin compatibility issues you will get alot of help.