We are finally (after 12 years!) getting to painting the last room in our home. It was originally painted with builder's paint, but being a "spare" room, we weren't in a rush to repaint. The room is on the 2nd floor, as are the other bedrooms, and there has never been any moisture or leakage problems. We purchased Kilz primer and it won't adhere to the ceiling or walls! It peels off as you apply it! What do you suggest? This is getting expensive and very frustrating. We had no issues with any of the other rooms.
Both answers so far are correct to a point. Older oil based enamels can be very difficult to cover. NOTE OF CAUTION, many older enamels contain LEAD and should not be sanded without first testing for lead and if positive, proper precautions taken. Lead testing kits can be purchased at most paint departments. Read the directions carefully, but they are really quite easy to use and are very accurate if used properly.
As mentioned, cleaning the walls is very important. Use TSP and scrub with something like Scotchbrite pads or green scrubbies. Use an oil based primer or sometimes pigmented shellac (Bulls Eye Primer) works well. If you still have an adhesion problem, there are liquid sandpaper prep liquids available that are good to use if you have lead based paint or high gloss finishes. They are kinda nasty to use, stink and are caustic, so follow the directions carefully and wear protective eye wear and gloves. If the surface tests negative for lead, sanding may be the best way to prep the surface. Using a DA sander with 100 to 150 grit paper, then cleaning the surface should do the trick. Be careful and Good luck.
You said you skim coated the surface and sanded. I assume you used joint compound because you sanded it. You need to get rid of the dust on the surface: So take a wet sponge mop and mop the surface, not too wet. Then I would rent a sprayer and spray the primer coat on.
Before applying any new paint on the walls it would be a good idea to wash down the walls with a hot water solution with TSP (tri-sodium phosphate) which should be available at your local hardware store. This will help remove any old oily grime and dirt on the walls that can keep paint from adhering. The TSP wash also helps remove some of the gloss surface of old paint which can also help with new paint adhesion.
To save effort initially you can try a small area before committing to the whole paint job. If you find that the TSP scrub down is not enough to fully solve the problem then you may want to lightly sand the old surface to roughen it up. I would sand before the TSP wash as the this will help to remove any remaining sanding residue from the wall.
Note that TSP solution is fairly caustic and can be quite hard on your hands and skin. Make sure to use some good quality "rubber" gloves that have cuffs that cover well up past your wrists.
I am going to throw this out there because I have run into it more than a few times with older homes. If you previous paint was an oil based paint then you need to get an oil based primer. You can go kilz or latex over that but you will need the first coat to be oil based primer.