Yes, we had numerous times where we were in an area alone, and some fairly surreal moments in the parks on my trip in October of 2001, since it was just after 9/11. Most every ride was just about a walk on, and there were times where they just allowed us to ride again. Although, the most surreal moment, wasn't actually at WDW, it was seeing Newark airport like pretty much empty. People didn't want to fly. I'll never forget how empty and quiet the airports were, I remember my luggage wheels were echoing so loudly over the tile floors while I was pulling the luggage. Not something you usually hear over the din of those normally bustling airports. Our record number of rides in a row that trip, actually happened at Universal though. We were on MIB 8x in a row.
We were the only ones there and we got all kinds of wonderful pictures and great interaction with the Mouse and his Lady. I remember thinking, "What do people mean about Disney being so crowded?" The pixie dust only lasted for about an hour, before it was as though the crowds appeared out of thin air, but by that time our family was hooked! We still had a fabulous time but our first experience at WDW is something we will never forget, no matter how many more times we go. 
So far it was a blast, but when I had to step into the library, I suddenly felt it getting eery. I was terrified that the CM would try to give me a good scare, was so happy just before the doors closed another family joined me.
