Q: Cascading Apple and other brand routers?

Bridge Mode is the correct setting for the AirPort....and the correct setting if you are trying to set up a Roaming network.

If you try to set up two separate networks, you will have a Double NAT error on the network.  Never a good thing, as it can cause a number of unpredictable things to occur on the network.  Things like communication issues between devices (which is likely why the AirPort is not working), slow browsing, difficulty reaching some web sites,....which will not load, or be very slow to load, and a number of other things.

Do you really want to try this?  You may already have your answer as far as whether this will work.

I've been trying to cascade a Linksys WRT54g and an Apple Airport Extreme router.  All the info I find tells me how to do it but it doesn't seem to allow the AirPort router to connect unless I set it to bridge mode.

Most of the information on getting these two to work was related to the much older 802.11g AirPort Extreme base station, and that, was to configure them for a Wireless Distribution System (WDS) type of extended network configuration. This is neither possible nor supported with the 802.11n or 802.11ac AirPort Extreme base stations.

Good catch, you bring up a good point and possibility.

I may have made the mistake of assuming that since the OP referenced a "Roaming" type of network in the title of his post, that this was in fact what he was asking about.   This type of network uses a wired Ethernet cable connection between devices....in this case the Linksys and AirPort.

If he is instead trying to use a wireless connection between these devices, then what is being asked is not going to work.

Hopefully, the OP will clarify whether he intends to connect the Linksys and AirPort using a wired Ethernet cable connection.....or.....whether he is hoping to use wireless as the connection between the devices.

I have it all set up now, however I'm also getting a DOUBLE NAT error and it all seems to still be working.  I can connect to the internet via both devices, just have to deal with the amber flashing light on the Apple router for now until something happens.

Thanks for the clarification. Sometimes you can get away with a Double NAT error on a simple network, but it can also result in some unpredictable problems on a network, which I listed above.  Things like Back to My Mac and online gaming won't work with a Double NAT.  If things seem to be OK for now, you can run things "as is".

If you are bothered by the flashing amber light, Apple does provide you with a way to "ignore" the Double NAT error, and the AirPort will display a green light.  In my opinion, this is a bit like placing a piece of black tape over the Check Engine light in your car to make a problem go away, but it's your call.

If you suspect that you might have a modem that is heading downhill, just connect a computer directly to the modem using an Ethernet cable for a few days and check the Internet connection from time to time.

If you notice that the connection is dropping, substitute a different Ethernet cable. If things continue to drop, then you likely have a bad modem.

If the modem is 4-5 years old, even if it seems to be OK,  it would be a good idea to replace it.