OCSP stapling and what web servers and browsers do in the face of errors
OCSP is an attempt to solve some of the problems of certificate revocation by having your browser or other TLS client check if TLS certificates are (still) valid when it sees them. OCSP stapling is an attempt to fix the privacy, performance, and infrastructure problems that OCSP creates by having the web server include ('staple') a sufficiently recent proof of its good OCSP certificate status in the TLS handshake; this proof is requested by the web server from its Certificate Authority's OCSP server every so often. As they say, so far so good. However, this raises two questions: what should the web server do if it can't get the OCSP proof from its CA when it asks, and what should browsers do in reaction to that?
It's possible to mark TLS certificates as 'OCSP must-staple', which means that the certificate is asserting that you should always see it with a stapled OCSP proof. If you're using such a certificate, it shouldn't matter what the web server does if it can't get an OCSP reply; no matter what, browsers should refuse to accept the certificate without it (whether they actually do is another matter, cf). However, most TLS certificates are not marked this way for various reasons (including that it's kind of dangerous because your CA can cut you off, either deliberately or accidentally).
For ordinary TLS certificates without OCSP must-staple, the web server has a choice of what to do in its conversation with the browser when it can't get a positive signed OCSP response from the CA. Either it can not include any OCSP stapling information at all, or it can pass on an OCSP error indication (or the signed OCSP response that says 'bad certificate' or 'unknown'). If the web server passes on OCSP errors to the browser, the browser may ignore them and continue with TLS as usual, ignore them and make its own OCSP query (even if it might not normally do an OCSP query), or report a TLS error and abort the connection.
(If the web server passes on a signed OCSP server response of 'bad certificate' or 'unknown', probably the browser should respect it.)
The safest thing for a web server to do is to only ever pass on a positive OCSP response. If your CA's OCSP server ever says anything other than 'your certificate is good', you don't say anything to the client (although if it says 'bad certificate' or 'unknown', you probably want to raise loud alarms in your monitoring system). In a perfect world, CA OCSP servers would always be operational and reliable, but in this world they aren't, so for most websites it's more likely that the CA's OCSP server has a problem than that you have a genuinely bad TLS certificate. When you're silent, at worst the browser will make its own OCSP query and get the same result, and at best either it won't query at all or it will query and get a good result.
As far as browsers go, Firefox will sometimes (or perhaps always) abort the TLS connection if it receives back certain OCSP errors, not just when it gets signed OCSP replies that contain bad statuses. For instance, if it receives an OCSP 'try later' status from the web server, it errors out with SEC_ERROR_OCSP_TRY_SERVER_LATER. This is not necessarily the greatest thing ever, because the 'try later' OCSP error is unsigned and so doesn't necessarily actually come from the CA's OCSP server. For that matter, the web server may decide to make up a 'try later' OCSP error status if it has some problem talking to the OCSP server. Chrome appears to ignore such 'try later' OCSP responses at least some of the time.
(Both web servers and browsers talk to OCSP servers over HTTP, not HTTPS, so they're vulnerable to man in the middle attacks for any responses that aren't signed. And of course the web server can give you any unsigned error status it wants to.)
PS: As far as OCSP must-staple goes, it appears that Cloudflare's blog doesn't use it any more, despite their 2017 blog entry pushing for it. Neither does Scott Helme. At this point I'd like to find a site that does use it just so I can check which of my tools actually report that.