The original question was, can you have a Backup DHCP server on the BDC? IF the PDC fails in any way, the BDC will takeover and so the BDC DHCP service will become sole provider of IP's.
Actually there is sort of a red herring here. There is no backup DHCP server. The DHCP service running on the BDC is always going to give out IP's to users, even if the PDC is up and running. Once it is set up and turned on it goes to work. In the instance of a failed PDC, it just keeps on giving out addresses as if nothing has happened.
It is a good idea/practice to have two DHCP servers when using DHCP; both capable of serving most, if not all, of the network. This helps with attaining the five 9's goal.
The big misunderstanding comes from Admins trying to run two DHCP servers with the same scope. This increases the chances of having duplicate IP's. The scopes need to be different.
For example. You have Class C address. You can have a max of 254 IP's within that class; but you only have 100 users. You can have the PDC DHCP server giving out 100 addresses--X.X.X.51 - X.X.X.150; and the BDC giving out a 100 addresses--X.X.X.151 - X.X.X.250. There is no overlap, no danger of duplicate IP's and each server is capable of handling the whole network if need be. This can also be adjusted based on admin/user needs.
A proper IP addressing scheme is paramount to the success of the network. There are many areas which can get a person into trouble and cause lingering problems.
There are many references out there on DHCP, and MS has a number of papers on this subject on their website. I would encourage anyone to read more on the subject before implementing this service.
Hope this helps
Bob