Short answer, "Yes, it's possible."
Most body shops tackle plastic car bumpers daily for cracks. The ones I've seen show from the backside the bonding "splice" bump, but you can't see it from the outside once they're done. Look for "Plastic Weld" kits in Google.
The brand "Bondo" as well as 3M (I think 3M owns Bondo now?) makes some sort of flexible plastic bumper patch kit, if I recall, and I used it on my car's bumper once. Take a look at the "3M Automix EZ Flexible Patch" kit. It worked well, just I had to spend quite a bit for the small amount of auto spray paint too but it was cheaper than the whole bumper by a lot! Sanding the outside can make a mess of the looks, but if you are good with a sprayer it shouldn't be too bad.
Only drawback is the amount of money for a good spray gun, compressor, supplies, and expensive auto paint and clear-coat to DIY. It adds up to a lot of cash and really quick. Sometimes, it's far easier to buy another panel even thought it costs $500+ or more, or let someone like a body shop handle it all and hopefully they'll be cheaper overall.