Can you have both sub-acute thyroiditis and hashimoto's thyroiditis?

Yes: Hashimoto's thyroiditis is considered a more chronic condition while subacute thyroiditis usually lasts for a few weeks or months and is probably caused by a variety of viruses. They can occur together but the diagnosis of subacute thyroiditis can be confounded by the simultaneous occurence of hashimoto's because the symptoms and the tests can be similar in some cases.

Yes: Sub acute simply refers to the time frame of when symptoms started. Acute usually means within a short period of time, chronic means a much longer time period, and sub acute is typically in between.