Andy did the same thing to Woody that Emily did to Jessie.

Although Woody didn't end up in a storage for decades like Jessie, Andy was a teenager in the 3rd movie.

Let's be real. Andy got older! So it's very likely that he paid less attention to his toys and more attention to Video Games, skateboarding, girls, his friends, rock/hip-hop music, just like any typical teenage boy.

Woody and the main toys implied that they haven't been played with for years! They were dumped and left in that toy chest.