GU
As I said, my rants previously about fellowships may not necessarily apply to me. I am doing what I want to do in a great place. But the fact remains is that the fellowship application process is a piece of **** unless you are lucky enough to have the right timing or other circumstances, or have you have a grant to study something directly related to the field of study. If you are the only person in your institution applying for the fellowship at your institution things will likely work out fine, provided you actually do have an interest in the field and have demonstrated it. But very often this is not the case, and many people try to do fellowships elsewhere or at the same time as someone else.
The lesson, in the current pathetic system we have, is to apply early, show interest in the area via projects, meeting attendance, good performance on the rotation, away electives if you like, etc. Unfortunately that often means you have to think about it before you are ready or know enough about the field you are applying in. Try to get early exposure in areas you are interested in, if your program allows it, even if it means doing extra work or sitting in on another rotation during a light period.
What it also means is that you are likely to be confronted with having to decide on a fellowship before another program you are interested in has decided. While again, sometimes this works out, many times it doesn't. Since everyone is on a different timeline, it can easily happen.
Bear in mind, if you plan to do private practice, you will have to compete for fellowships with people who want to do academics (or at least say they do), and they will have a leg up at many programs. If you decide to do academics, things are more straightforward but there are a lot of people who overrepresent their own interest in academics for the purposes of landing a fellowship (a sad but true reality, in which people feel they can't be honest). Part of this though is a consequence of having to apply to early - if the fellowship is still two years away, are you really sure about the exact course you want your career to take? I wasn't, although I am more certain now.
I think perhaps, a lot of people do extra fellowships who don't need them. If you train at a good residency program and want to go into private practice, one fellowship is almost always enough to get a good job. I have heard many attendings say that these days it seems like people are doing more fellowships, but that they don't know why because it doesn't really make them a better candidate.