Ever see a spike in the 404 errors reported in Google Search Console's crawl error report? Especially from URLs that have been 404ed for ages? Well, it can happen said Google's John Mueller. He said that Google will periodically retry old 404 URLs to see if anything changed with those URLs.
He posted a response on Twitter saying:
@davidwkelley @googlewmc AFAIK no change in reporting. We periodically recrawl older URLs to see if there's something awesome there, maybe you're seeing that.
— John ☆.o(≧▽≦)o.☆ (@JohnMu) May 9, 2017
So if you see such spikes, there is no need to panic, espesially if those pages should be 404ing. If you see 404 responses on pages that should exist, then you probably should panic. Otherwise, don't worry about it.
It is a good thing that Google retries old 404 URLs. I mean, what if you put new content on a URL that once was 404ed, don't you want Google to find it?
Forum discussion at Twitter.