I have those same cabinets in my 3 bathrooms, and I've seriously been thinking about painting the one in the master bath. I really want to put a wall cabinet in there, and I can't find anything to match the existing light oak vanity.
I have painted cabinets before, but they were old circa 1970's kitchen cabinets that were in my previous house. In order to insure a good paint adhesion on a high use item, you need to do a good prep job. First of all, remove the doors and drawers. Using a small piece of tape, label what goes where. Take off the hinges and place them in individual zip-lock bags. Again, label them carefully. Take off the knobs/handles. If there is any chance of grease residue on the cabinets, you should wash them down with TSP. (Not likely for bathroom cabinets.) Next, lightly sand to rough up the finish. If you have a power sander use it, but practice first on a scrap piece of wood until you get a feel for it. You might need to hand sand some of the weird shaped areas. Carefully clean off all sanding dust, trying not to get the wood too wet. Tack cloths work good for this too. Let dry overnight. Prime with a latex based wood primer. If you plan on using a dark color paint, you can get the primer tinted a darker color. Let dry overnight. Follow up with 2 to 3 coats of paint, letting each coat dry overnight. It's much easier to do a better job on the doors by laying them flat. Just be careful of drips traveling to the opposite side. I can't recall the brand, but I used a kitchen/bath cabinet latex paint from Lowe's or Home Depot. I was able to get it tinted to the color that I wanted. (I did my cabinets in a cream shade.) Take the opportunity of changing the knobs/handles to something more contemporary.
Of course, you can take shortcuts (like not sanding or priming), but if it's a high use item you might end up with chipping and scratching. I used my kitchen cabinets for almost 5 years before we sold the house, and they still looked perfect. When the eventual buyers toured the house, they commented to the realtor about the beautiful NEW cabinets. If only they'd saw the awful 1970's cabinets that were hiding behind my great paint job!