There was a story here a couple of weeks ago or something with a post mortem from someone who got their websites taken offline and access to their mail suspended or taken offline because some automated system at a giant corporation decided so.
It stayed offline for hours because there was no competent humans available with authority to override the clearly (AFAIK) stupid decision made by the system.
Personally a key takeaway from that (except not depending on said company) was to make sure one has at least two domains, preferably with two registrars, and make sure user generated content is not available on the one you depend on for email etc.
Note that in that case the user content was on a separate domain—the registrar blocked the entire account.
(Although I think there might have been some comments about user generated content on the main domain.)
EDIT: I couldn't find anything directly on point, but I did find this related horror story about a company whose ICANN email contact information no longer worked because the company's contact person had left the company. An ICANN email to the contact person bounced back. ICANN took the entire domain offline. The company's Web person had a lot of trouble getting through to a human at ICANN to straighten things out — and even then: "To make a long story short, my client was required to submit articles of incorporation, bank statements and other documents in order to get the domain working again." 
EDIT 2: Thanks to Chris Morgan, who provided a link to a GitBook horror story. 
I added both stories to my "Startup Law 101" page , with a hat tip to both Erik and Chris.