Is it impossible? It depends on how deep they are.

If the scratches are very light and are just in the clearcoat or basecoat (still looks body-color, not plastic-bumper-color) then a polishing compound can take the edge off the scratch and make it invisible, or at least less noticeable.

If the scratch is deep enough to go through the paint to the plastic, then the next step would be to try touch-up paint. You can get a small bottle to match your car from your dealership's parts dept. This is applied with a small brush (attached to the cap in the bottle) and is going to be somewhat noticeable, but better than the lack of paint. Polishing or rubbing compound after the touch-up paint has dried can help make the spot blend in. Or, you can try products like Langka (search these forums for 'langka' and you should find some discussion on this product).

A body shop can take the bumper off your car, sand it down, respray the entire bumper, and put it back on your car. This should look good-as-new (if it doesn't, make them fix it -- as you're probably looking at around $2-300 just to do the bumper). You don't actually have to replace the bumper unless it's cracked or split or dented (as mine was when the Nissan pickup behind me used it as a deceleration aide). To replace and repaint the bumper, you're looking at around $600.

Whichever route you choose, you'll need to re-wax the location. If you just polished the spot, you can wax immediately. If there's been any painting, give it a few weeks to fully cure before waxing.

2003.5 G35 Sedan Desert Platinum/Graphite Premium/Sport/Aero/Nav/Winter

 

My bumper has a crack/cut in it about 1-2 inches. All the body shops that I called said they had to replace the back bumper. The cost was in the range of $700-$1100 (3 different estimates to replace and paint a new bumper). My dealer (in San Diego) said that the crack can be filled, sanded and repainted for about $200. They said that it would look like new. I will probably do the $200 job and see how it turns out...

ETicketRide
2003 G35 Sedan/Ivory Pearl/Willow Premium/Sport/Tint

 

That seems kind of high. I'm in San Diego, too, and I had my rear bumper cover replaced and repainted for between $600 and $700. (At the expense of the other guy's insurance, of course!) And it's an OEM bumper cover, not an aftermarket bumper cover.

2003.5 G35 Sedan Desert Platinum/Graphite Premium/Sport/Aero/Nav/Winter

 

Ryoken-
Where did you take the car? Were you happy with the work? Thanks...

ETicketRide
2003 G35 Sedan/Ivory Pearl/Willow Premium/Sport/Tint

 

Here's the whole story if you want:
Time to start a new club...

All-American Paint & Body, on Buckhorn St. (Near 163/Clairmont Mesa Blvd)
I've used them twice, the first time was on a Grand Cherokee, also needing a rear bumper cover. Yes, I was happy with the work they did, both times. My experience with quality of work and customer service from them have both been excellent. They also happen to be on the pre-approved list for most insurance companies, which means you don't have to wait for an adjustor to double-check their estimates before they can start repairing.

2003.5 G35 Sedan Desert Platinum/Graphite Premium/Sport/Aero/Nav/Winter