Can I take aspirin and dipyridamole?

Yes but bleeding ris. The combination exists in one pill called " aggrenox". It is useful in stroke prevention, but it can cause bleeding, and only a doctor can decide if the benefits outweigh the risks of taking both.
Yes. There is actually a commercially available drug which combines these medications. Google the term "aggrenox".

Related Questions

Does dipyridamole add to aspirin in secondary prevention of stroke?

Yes. It is called aggrenox. It is used for patients who have tia, transient ischemic attack or stroke on aspirin, and there is less bleeding than with plavix (clopidogrel).
Yes. Yes. Other important factors are BP & cholesterol control.

Why is my partial thromboplastin time in the lowest range even though I'm taking aspirin and dipyridamole?

Unrelated. Partial thromboplastin time relates to factors involved in clotting of blood that are unrelated to platelets. This is a very common point of confusion. This will help you get started; read up further if you like.