Is it possible to completely brick your Android phone by using a Custom ROM? I understand that you can completely "brick" (render unusuable) your iPhone by attempting to Jailbreak incorrectly.

Whether or not this is actually the case, is it possible to do the same with your Android phone?  Or is the nature of that operating system and the handsets such that you can always install a new ROM on it if everything goes pear-shaped?

I assume there could be a difference due to how closely the iPhone is tied to iTunes, compared to how independent Android phones seem.
Yes it is possible to brick and that is why all root methods and custom roms give warnings on their threads. But that said, it is very hard for someone without much knowledge to "brick" a device as long as you follow the directions given. HTC does a great job on their devices and makes it difficult to brick them. Almost always there is a fail safe of some kind.

Most people that claim they bricked their device are mis-using the term and just throwing it around. Even someone that completely cracked their screen and could not use the screen of the phone was still able to successfully root and pull off needed files. The brick cries are no different now then they were with Windows Mobile devices that people tried to unlock. Every time someone saw the bootloader screen they cried brick, but just didn't understand that screen was normal.

If you want more reading on the topic, check out XDA Forums. Also I wrote a guide on XDA forums for rooting and backing up information, tips, etc. While the article was specifically written for the Evo 4g, most of the information still applies to any Android phone. I recommend checking it out for more info.