I had to move several of my high-uptime-requirement WP sites off of DreamHost and onto WPEngine because of this issue more than a year ago.
What was happening then (and probably still is) per the DreamHost support techs is that there is a temporary directory on the shared hosting environment(s) that is being filled up and is not automatically emptied, causing the sites to be completely unresponsive. I was calling DH every few days during prime restaurant hours (several of my clients are restaurants with online ordering) and begging them to clear the temp directory without having to wait for them to suggest 5 other things that were NOT the problem. It’s not that they weren’t being helpful, it’s that every time I called the tech on duty would have to re-troubleshoot it because I was just some guy telling them what to do…
I am pretty sure that DreamHost DOES have dedicated WordPress hosting, which is more solid than just throwing WP installs onto their standard non-specialized plan (aka, the one I pay $10 monthly for and host like 30 sites on…) — so you definitely get what you pay for.
Move critical sites over to a WP-specific plan, or pay the big bucks, go with WPEngine and never have to think about it again.