Why can I login to a box even if the AD Domain controller is down?

Solution 1:

By default, Windows will cache the last 10-25 users to log into a machine (depending on OS version). This behavior is configurable via GPO and is commonly turned off completely in instances where security is critical.

If you tried to log into a workstation or member server that you had never logged into while all of your DCs are unreachable, you would get an error stating There are currently no logon servers available to service the logon request

Solution 2:

Yes, your credentials are cached on each machine you log in to. If you had not logged in to a given machine before the DC went down, you would not be able to log in because your credentials would not be available.