Similar issue with me. My wife's MacAir is working fine whilst my iMac isn't. Both are sharing the same Office 365. Both machines are a month old and have been working fine to date.
I logged a call with Tech support (after having to install Firefox because Safari and Chrome couldn't deal with Microsoft's help desk :-() They logged my call and promised to call me back tomorrow between 9 and 11am. Not easy to run a business without e-mail
access, Word and Excel these days... maybe it's time to give Pages and Numbers a go...
I did get my Tech call from Microsoft this morning as promised. Thankfully I had already checked and the issue had been resolved prior to the call.
As it turns out it had nothing to do with me or my computer so I didn't have to worry about re-installing OS X as some people tried to do, or stand on my head with an apple in each hand while reciting the ode to Kublah Khan!
My feedback to the tech person (he was very patient and polite, I have nothing but good things to say about him) was as follows;
Next time there is an issue that affects a lot of users (Office 365 or IE or Windows or any other application) be pro-active. Send out an SMS that acknowledges there is a problem and you are taking steps to fix it. Something like "Some Office 365 users are currently experiencing problems with authenticating their software. We are currently working to resolve the issue and we will update you when we have resolved it".
This approach would have done a number of things including unclogging Microsoft's help lines, letting users know that others are having the same issue and someone is actually working to fix it.
Instead of irate people wasting time and effort looking for fixes and trying to do things that don't ultimately help you could end up with 'valuable clients' that are actually feeling like someone is looking out for them.
Hmm, wait, I think I may have landed on the wrong planet!