Can a single SQL Server instance issue the same SYSDATETIME twice?
Yes. The resolution of the Windows clock is decent, but not perfect. You should use another field, such as an INT IDENTITY, to ensure uniqueness.  But go ahead and cluster on the date first, if it suits your needs.

Alternatively, you could use a DATETIME PK, but supervise inserts with a stored procedure that verifies the uniqueness of the incoming date, and pauses if necessary.  I can whip up an example if you'd like.

Edit:

Another option, borrowing from ypercube's comment, would be a non-unique clustered index on the date field, and a unique non-clustered primary key on the ID field. This would keep your other indices smaller, as they would point to the narrow ID column, but would also entail two hops to get to actual data.