My work firewall for wired internet blocks almost every port except http, https and maybe a few other web-related ports (993, 465 and everything else to do with IMAP is blocked). Impenetrable for, among others, Gmail. My office mail is via Exchange/MAPI (the mail server is also a web server? strange) and is not stopped by the firewall.
I also have wireless work WIFI account, which of course is slower than the wired connection, but is not blocked by any firewall.
When the wired internet is connected, a simultaneous check of my email accounts gives a message for Gmail: "could not connect to mail server #.#.####@#####.###; the connection was refused." I understand why.
The funny thing is that if I disable the wired internet, the simultaneous check for both my e-mail accounts completes successfully! This would not be necessary if checking Gmail would use the (still available) wireless connection!
Would it somehow be possible to couple the gmail account to the wireless internet connection and the other internet traffic to the wired internet connection? Then it would be possible to keep reading Gmail in an e-mail program (e.g. Thunderbird) instead of having to use WebMail.