Yes you can - I have this setup at home, one upstairs one downstairs.
Caveat though, the second one (not directly connected to the Internet) either needs to be put in "wireless access point" mode, if possible, or will need some settings changed for the whole thing to work:
The second router will need DHCP disabling, its IP address setting to one far enough up the scale so it doesn't cause issues (e.g. 192.168.0.200) and its default gateway setting to the IP address of the first one (e.g. 192.168.0.1). If you don't disable DHCP then you get two routers trying to hand out network settings and nothing will work, and if you don't set the gateway then the 2nd router won't know where to send Internet traffic.
If the 2nd router has an "access point" setting you just use that and everything is done for you.
Very handy for keeping the kids connection separate and disableable at will (automatically on ours)!