One time I had to rush at a customer's site whose quite impressive Exchange 2010 system (talking about > 20 servers here) kept randomly stopping mail delivery, which definitely isn't something you would like a mail system to do.
The root issue was a random crash of the main mail transport service (edgetransport.exe, for those who know and/or care) on all five of the Internet-facing servers.
The root issue of that... the whole setup was stuck on O.S. and Exchange releases from more than a year and a half before, because after having it designed and installed by Really Expensive Consultants, nobody bothered patching anything, ever. Even if the servers themselves were screaming for that, since they had Internet access and were perfectly able to yell "hey, I need updates, please do them!". This included a missing Windows service pack and all subsequent updates, a missing Exchange service pack and subsequent updates, and even updates for the antimalware software running on those servers which were actually experiencing the problem. I'm quite sure that, had the servers been physical instead of virtual, firmwares and drivers would have been treated in the exact same way. I don't want to know how their real, physical servers were managed (or, more likely, left to themselves).
First thing I did, was giving the customary "please update everything to the latest releases, then we'll talk" speech. That wasn't enough. They wanted to know for sure if that was going to really fix their problem, or they weren't willing to touch anything. Even if their system was at that stage performing no useful function at all, because it just kept randomly ceasing to work. The fact they were running software not even anymore supported by the vendor itself was just that irrelevant to them.
Ok.
Proof they asked for, proof they got: I spent a whole (of course conspicuously billed) afternoon on MSDN, and found at least five different Exchange updates, which of course their servers were all lacking, which had in their release notes "this fixes random crashes of the Edge Transport Service".
They hadn't even bothered with 5 minutes of Google, they just kept restarting those servers and calling more and more consultants which, it seems, were even more clueless than them. And when someone finally told them "maybe you really should run more up-to-date software", they just didn't listen.
Oh, and the icing on the cake is, this was definitely not some cheap understaffed shop. This was Italy's biggest telecommunications company, which was hosting this Exchange system for a government agency (and not a small one).