I had two specific problems with running late-gen PS2 games on the original model of the console that I've never heard of anyone else having:

1. In God of War 2, at the end of the grappling scene (https://www.youtube.com/watch?v=vRajEYd47tc), Kratos would bust through the column and land in complete blackness. There were no textures and no models, and I couldn't walk anywhere. this happened consistently for a week, until one day it just happened to work.
2. In Ratchet and Clank: Up Your Arsenal, during one of the between-planet loads where you're shown a montage of R+C flying through space, the planet never loaded, and just came scene after scene of the two flying through space. Again, this happened consistently for a week until one day I said fuck it and left it running to see what would happen. After at least an hour the planet finally loaded.

Anyone else have any problems like these? I would hesitate to say this was related to the hardware except for the fact that these loading problems were repeatable in completely different games.

The original PS2 had so many revisions so you might want to look into your specific model. Played GoW2 on a fat PS2 and had no issues. Can't help w/ Ratchet.

@m2thek: I never had any issues with my fat, however, my brother did have an issue with the last version of the PS2 slim and backward compatibility with a few PS1 games. So be careful if you get one of the 70000-series slims.

When I bought the Greatest Hits version of Gran Turismo 4, it would not run on my fat PS2. It didn't matter how many copies of the game I tried, they all refused to start.