I'm a bit late to this discussion, but perhaps can add something from a residency viewpoint. I've had several residents with disabilities, and I've tried to reasonably accommodate all of them. Some more successfully than others.
The ADA is a complicated law, and the reference at the top of this thread (in the first few posts) is a nice summary as to how it might affect residency programs. In the law, like medicine, there are few absolutes -- so it's impossible to say with 100% certainty what's legal, or required, by the ADA in any specific situation. However, a few generalities may be helpful in helping you think this through.
First, let's not lose sight of the purpose of the ADA -- to allow disabled individuals, with reasonable accommodations, to perform at the same level as non-disabled individuals. So, if you're a cashier and you have a back problem, and sitting on a stool fixes that problem, and having you sit on a stool doesn't create any huge hardship for me as an employer, then I have to accommodate that request. Where it starts to get sticky is when your accommodation is a request to change the nature of your job.
In general, the ADA is much more expansive in a schooling situation. A medical school might be required to decrease the number of classes you take per semester, for example -- assuming that the school doesn't use an integrated curriculum where this becomes impossible. During your clinical years they might decrease your clinical workload, give you more days off, etc -- all because as a student you're not fulfilling any essential function, you're just there to learn. If it takes you longer to learn the material at a slower pace, that's a completely reasonable accommodation.
When you come to residency, the rules change. The ADA becomes more strict, because now as an employee you're performing a service for your employer. Much of the conversation revolves around a single issue -- does the requested accommodation conflict with the essential functions of the job? If so, by requesting the accommodation you make yourself ineligible for the position, so any ADA protection ends. That is the point of many of the examples in the link on the first page. This is a very complicated concept, so perhaps some examples are in order.
Let's look at your situation. First, let's define what a typical residency work schedule would look like. You would have "busy" months and "elective" months. On elective, you'd work from 8-5 M-F with most weekends off. Let's assume that's a workable schedule for you without modification. On "Busy" months, you'd work 60-70 hour weeks, have 1 day off per weekend on average. You might work both weekend days one week, and then have 2 days off another, so that would be working 12 days in a row. You might also have a mixture of day and night shifts -- perhaps weeks with 5 night shifts in a row (often called "night float"). Although busy and elective months would be spread out as much as possible, it's likely the number of busy months might be much more than elective ones (especially early in training), so you should expect to do several busy months in a row. This is very vague, but is probably typical for many residency programs esp in psych. Remember that four months of psych internship or so is usually done in IM.
You have some illness which makes you fatigued, such that you do not think you can work a full schedule. So, what accommodations do you suggest would work? I see several options:
1. You get scheduled for a usual full schedule, and any time you feel fatigued you call in "sick". This is not acceptable. The ADA considers showing up for work regularly and reliably an essential function, esp in healthcare where a regular and reliable staffing is critical to patient safety.
2. You request to have "part time" work where your hours are 50% all the time. If training was simply morning and afternoon clinic, and then you'd only have morning clinic, this might work. But when you're working on the inpatient service, you can't just work 1/2 a day. This isn't menial labor that someone else can pick up what you've left off. You need to be present for your whole shift, and I can't redefine the shifts around your needs.
3. You request "part time" work where you work at 100% speed for 2 weeks, then have 2 weeks off. This is much more reasonable than options #1 and #2. I see two major problems -- I'd need to find someone else willing to work the other 50% (which is possible but not likely), and I'm not certain that this would actually address your disability (although that may be incorrect).
4. You request no night shifts. This is a non starter. Night shifts are a curricular part of training. medicine at night is different from medicine during the day. You can't just miss nights.
5. You work 100% during the week. But you need time off to recover, perhaps both weekend days each weekend. You're willing to swap -- so on elective months you'll work some busy weekends, to give yourself some additional rest time during the busy months. This is closer to a reasonable option. However, I still think it's problematic for several reasons, mainly one of continuity. It's better for patients if the same people care for them over an extended period of time. So, if you make more coverage swaps in the schedule, this can create problems for patients and your coworkers. All that said, this is probably the best solution of all of those I've detailed, and would have some chance of passing legal muster. Note that it still gives you 12 workdays in a row when on your electives, and if busy months outnumber electives you simply don't have enough weekends to make it work.
Perhaps I'm missing some other option, I'm happy to discuss. Several other problems with this plan:
A. Residency programs have strict caps on the number of residents allowed at all times. If you were to complete a residency at 50% speed, unless I have a matching resident at 50% I'm always going to be short workers. The ADA does not require employers to hire new people to do the work that someone with a disability can't.
B. Residency contracts are one year at a time. If I'm not happy with your performance, I can simply decide not to offer you a contract for the next year. I'm not certain whether the ADA protects you from this or not. Perhaps there's an "implied" contract for your second year. But maybe not.
C. If you end up in a disagreement with your employer about all of this, they will simply fire you and then plan to fight it out in the courts. And you've mentioned above that you don't want that, and courts take forever to settle things.
So, bottom line:
1. Getting a spot with limits on how much you're able to work will greatly limit your ability to obtain a position.
2. Going to a top notch program, being a star performer, and applying to less competitive / more flexible programs will increase your chances of success.
3. Although it's legal to hide your disability until you're hired, there is a price to pay for that. If your employer feels they cannot (and do not have to) accommodate your disability because it would alter the essential functions of the position, you will be let go. You'll then need to sue, or to reapply. Alternatively, you could be 100% open about the situation -- in that case you'll presumably get less interest/offers, but programs that do consider you know what they are getting into and will be much more willing to work with you on the issue. Also, if you're open about the situation and someone takes you, they then will have a much harder time convincing a court that they cannot accommodate your disability -- since they hired you with full knowledge of the situation.