Just seen this. On 3rd December I received a letter from DVLA to inform me that the Direct Debit for my Tax had not been taken out and so, my car was not taxed. The Road Tax should have been taken out on 1st of October ! So I had been driving around for 2 months, blissfully unaware that my car was not taxed. The letter blamed the oversight on "a system failure" and advised me to tax my car ASAP which I did. They did at least apologise for the error, but , when I phoned the automated system to tax my car, I was warned that as I had let the tax run out, I might be liable for prosecution. What a cheek! I hadn't noticed that the Direct Debit had not been taken out, but it took DVLA 2 months to spot the error. I assume it hadn't flagged up on the system, otherwise I could have had Plod pulling me over.
As the OP suggests, don't rely on DVLA getting it right all the time.
Paul