That's the thing...at many programs (especially the top-tier ones), it can be hard to bust into a fellowship from the outside since they can easily fill with internal candidates. One notable exception to this is Harvard Dermpath (and probably other dermpath fellowships out there) which takes applications from all over the country and doesn't give necessarily much preference to internal candidates from MGH, BWH, and BI. If you're interested in a fellowship at an outside program, I would inquire early but not expect to get a sure yes/no answer because they have to wait to see if they will fill with internal folks (who usually get first dibs).
Cyto fellowships, from what I hear, are popular. It's a boarded specialty, gives you a marketable skill, and your work hours are not likely to be too killer. So yeah, the one cyto position at Stanford is probably difficult to snag.