Yes, it is possible, but with some caveats. First of all, I would highly recommend a mountain bike, should you decide to use the off-road trails. There are plenty of roads from SJPDP to Santiago, and a road bike should work well there, should you really want to use a touring bike.
Jane and I rode our touring bikes from Barcelona to Logroño last summer (2013) and then onto Leon, in all, 800 km. She suffered a bad crash in Leon and needed a bunch of stitches in and on her knee, so that was the end of the bike ride. She was able to walk fine, so we shipped the bikes home and finished walking from Lisbon, Portugal, to Santiago instead. We ran into a German pilgrim there that had just started walking her Camino a few days before and was whacked by a passing bicyclist. Both were injured, but she got the worst of it and needed stitches on her face and was very sore. She bravely returned to the Camino after her medical attention, it had to be a difficult decision.
Courteous bike riders are no problem, but it seems that many that ride the Camino know nothing of how to approach and pass pedestrians. Many fly by, with no warning and the scared pedestrians jump to get out of the way, and 50% of the time, jump into the path of the rider. I found it better to not move to the right or left when I heard a bike coming and just let them work around me. If there was time to turn and look and get out of the way properly, I would. The ones that ring their bells (those that actually have them) and give plenty of warning are no problem, the others are a hazard. There is room for all, but there needs to be common sense, which sometimes is in short supply.
We found, at least from Barcelona to Logroño, that the roads were narrow and the truck traffic was very heavy. To their credit, the drivers were decent enough, but the narrow roads made it difficult for all concerned. If I were to do it again, I'd take a mountain bike and ride the off-road routes. Spain's roads are not conducive to touring bike riding. Interestingly, I found the Portuguese drivers to be much worse and not very courteous to pedestrians, and yet the people there were wonderful. It seemed the drivers have not yet discovered the brake pedal.
It's not a race, whether walking or riding, take the time to take in the sights, and use some caution and all should go well.