No, the skill sets are not too separate at all. As others have indicated knowledge of AD, group policy, authentication protocols, programming, the fundamentals of database design and querying are all important and make you both more effective as a DBA and more marketable. Also, ignore your knowledge of storage, it's configuration, and performance at your peril. In my experience, storage is often ignored and often a crippling bottleneck.
Knowledge of the mechanics of the client-server application paradigm (if you will) help make you more effective in support of your employers resources. For instance, being able to advise the app team that looping over a query vs. querying all the rows they need at once is generally a poor idea and why. Even something as simple as advising on the configuration of a connection string can be of tremendous value.
You also need to be able to explain to Sr. System Admins, storage admins, etc., in clear terms what your db server needs in terms of resources and configuration and why. Knowledge in these areas only aids you in this. For example, in many cases, the storage team will not know how to configure disks for SQL Server. You need to be able to tell them how and justify it.
When interviewing at my current employer the statement that "you're also a Windows Admin" was explicitly made in terms of a benefit to the organization. That skill (and many others) only make you more valuable and certainly do not diminish you or are a waste of your time.