I've had a whole carriage twice and a whole train once, though 2 of them are combined.
The first time was in the early 1980's, on what was the rump of "The Clansman". It used to run Euston - Birmingham - Crewe - Preston - Carlisle - Motherwell - Coatbridge - Stirling - Inverness. Then it got re-routed via Edinburgh. Shortly after that, the principal service to Inverness got switched to Kings Cross, BUT they left the Euston - Edinburgh part of it in place for about a year. I got on it once at Carlisle and had the entire carriage to myself all the way to Edinburgh!
The second time was in 2002, give or take a year. It was a Sunday, and my train from Edinburgh got delayed at York for 3 hours when it ripped down the OHLE near Northallerton. We got pulled into York with a Thunderbird and had to get a much later train that got me into Kings Cross just after midnight. I was bricking it, as I still had to get to Stanmore and I was worried in case the Underground closed down. I got a Met tube train from Kings Cross to Wembley Park, then managed to get a Jubilee Line train to Stanmore. I was near the back of the train, and I had the carriage all to myself. Although we stopped at every station, I had no idea if people were getting on and/or off, but when I arrived at Stanmore, I noticed I was the only person walking down the platform to the ticket gate. So I had the train to myself from AT LEAST Canon's Park to Stanmore!