How common is it to get a thrombosis in your arm after a routine blood draw?

Common. The vein that is punctured for the test will often clot and form a lump of sorts.-- especially if the phlebotomist missed vein or went thru it with the needle. This is a superficial vein, usually not serious and responds to heat and rest. If there is redness or persistent pain, see and md. Not all clots are serious but, when in doubt , of course please please see an md feel better.