The paneling should have been sanded well, and primed with a bonding sealing type primer, and then 2 coats of whatever paint you are using. It will keep peeling, so best thing now, is to sand it and feather it out where it is peeling, and whatever comes off when you are sanding, then get the bonding sealing primer on it, and then repaint. Its a common problem, these days, and one you shouldnt expect from a painting contractor. I had the same problem, here with all the trim in the house. It took forever to get it all redone.

In answer to questions, it was a dark wood panel, stained but not glossy or waxy. Here is a picture of it after it was painted with Benjamin Moore Regal Select, pearl finish.

Of course it needed prep. You have no idea what was used on it before- old English and pledge both are notorious for making the surface resistant to paint later. There is nothing to be done topically to readhere what’s there- scrape loose areas and mechanically sand with 220 grit paper. Shellac prime. Scrape with your nail to make sure it’s adhered. Then proceed with paint. Frankly the original painters should do it with there tails between their legs.

*their. He says with his tail between his legs.

Here is a picture of it after it was painted

How about one or two showing where it's peeling?