OK, thanks for the information. The issue is that the WAN port on the AirPort is not negotiating things correctly with the modem.
Unfortunately, Apple removed the settings that could be used to manually adjust things on the WAN "O" port about 3 years ago, so we are all stuck with an "Automatic" WAN port setting that cannot be adjusted. It's great when it works, which is most of the time, but there are only a limited number of things that you can do when it doesn't.
Some users.....including me....have reported that inserting a Ethernet switch between the modem and the AirPort will sometimes help things. What is happening here is that the switch does a better job of negotiating with the modem, then sends the Ethernet signal on to the AirPort router.
If you decide to try this, look for a 5-port Gigabit Ethernet switch, and ask the store if they have a 7-14 day return privilege in case the switch does not help.
I've seen a few reports from users that a crossover type Ethernet cable between the modem and AirPort also helped. So, you could try to replace the existing Ethernet cable connection between the modem and AirPort with a "crossover cable" to see if that helps.
Other than that, you could try resetting the AirPort back to factory default settings and then set it up again. This is a lot of work....and frankly does not usually help.....so that would be a last resort.
The AirPort Time Capsule has a 1 year warranty in the U.S., but if you have AppleCare in effect on one of your Macs, the TC would be covered by that protection. If nothing that you have tried solves the issue, then I would ask Apple to replace the AirPort and hope for better luck on the next product.