In spite of the words from the SkyTeam website:
You only check-in once when connecting to other SkyTeam airline flights. With a maximum of two SkyTeam airlines in your itinerary, and a maximum of three connections.
Air France's computers can't connect your bags more than twice.
Had an F award ticket routing LHR-CDG-IAH-SLC-SEA (yes, you do take those routings to get a seat on the day you want in the class you want). At LHR, the bags got ticketed to SLC. Since I had a 40 minute connect there, I knew I couldn't claim them and still make my connect. The GA said that their systems would not allow for a 4 flight itinerary for the bag tags.
Since they weren't going all the way to SEA, I had them tag them to IAH (where I would have to clear customs anyhow). No problem. However, at IAH, CO would not retag the bags for the next two DL flights, and there was no one from DL at the international recheck after customs. The only option was for us to schlep the bags from terminal E to terminal A on our own, with no assist from anyone.
So, beware if you have a multi-leg itinerary and have to start it on AF!!