If everything else works and one site does not, the connection is fine. The question becomes where the request is failing: at the name lookup, at the site itself, or at something in between such as a cached copy of an old page.
Name lookup problems produce a distinctive symptom: the page never starts loading and the error mentions the address rather than the content. Switching to a different name service or another network will show whether the lookup is the issue.
Certificate warnings mean the secure connection could not be verified. Most often the device clock is wrong, which makes valid certificates appear expired, and correcting the time fixes every site at once.
Cached content causes the confusing cases. The browser holds an old copy of a page or a login token, and the site fails in ways that make no sense until the cache is cleared or a private window is used.