Q: I'm trying to connect to SQL Server via SQL Server Management Studio on the local SQL Server machine. Even though I'm a systems admin on the machine, the logon fails; why?
A: Remember that Windows uses User Account Control (UAC); when you're logged on, you actually get two tokens—a regular user token and an administrator token. If you start SQL Server Management Studio (SSMS) normally, it runs as your user privilege token, which means your group memberships (e.g., builtin\administrators, which contains Domain Admins) won't apply and your logon will fail.
Therefore, you should run SSMS in elevated mode (right-click, and select Run as administrator). It will then run as your administrator token, and you'll be able to log on.