How do I login to SQL Server via group membership in another domain
In my case I was able to narrow down my trouble by working with my sysadmin about how the group was defined.

Knowing that the SQL Server was hosted on a machine that had joined another domain and was running with a user account on that other domain, we looked at the Group Scopes in active directory.

My user could authenticate properly after the scope was changed to universal. 

If I find an authoritative source on SQL Server's behavior with respect to these domain group types, I'll be sure to update this answer.