Can peeling laminate kitchen cupboards be painted?

You can paint them but you'll want to sand them down first

Yes, once you remove the laminate sand the cupboards prior to priming/painting, you should get good paint adherence that way

I would beware of attempting to remove the laminate. It's been securely glued to a particle board piece. First, it's unlikely you will be able to peel away all the laminate. It's more likely to snap where the glue is still intact, leaving you with a rough edge involving 2 different surface materials.

Second, particle board can break down easily. It's made of wood shavings mixed with a glue. Any moisture that's gotten into it will cause it to "shred" while trying to paint. I think it may be best to attempt to reglue the loosened areas and then paint them when intact.

Yes, you can. Any part that is already peeling, just peel it off. Make sure to sand gently and seal that part with a couple coats of sealing primer, so that water cannot get into the compressed wood. Then you can paint the whole thing. Leave the rest of it on. Peel off only the broken ones.

So, start with cleaning/washing all the plastic coated cabinets really well.

Then hand sand (with medium grit sanding sponge) them in the direction that the grain would go. Clean that off.

Wipe them down with white rubbing alcohol.

I recommend PPG/Porter Paints Advantage 900 paint, in semi-gloss sheen. It is very, very durable. (Soap and water clean up)

Buy VERY good quality, soft brushes for acrylic paint.

Put thin coats on, and let each coat dry overnight.

In the end, once the color has built up sufficiently, you will not need to put a sealing poly coat on.

Just general FYI: Paint takes 30 days to cure. So try not to touch your cabinets too much for at least a week after you finish painting.

IF your weather is wet for very long, then you will need even longer before you use your kitchen fully.