This is a very technical issue that there are many different viewpoints on. Overlapping heat affected zones can cause some major problems if not taken care of properly. Under ASME Sec VIII Div 1, UW-14, you are actually allowed to have weld-on-weld connections. This can cause some major issues regarding hardnesses and strengths of the final product, and additional testing needs to be done.
Failure modes? I have heard my boss talk about some testing that was done that had hardnesses all over the map. So I would guess that the problem is either going to be a loss of strength or brittle fracture depending upon where the hardness ended up.
To be perfectly safe, you could probably qualify a weld-on-weld procedure that would prove that you had adequate mechanical properties in the finished product. I highly recommend that you check with a welding specialist if you plan on doing welds very close together.
As a rule of thumb, we typically do not spend a lot of time worrying about the distance between welds as long as they are at least:
1) 4" apart; or
2) 4 times the thickness of the weld apart
(whichever is greater)
Hope there was something useful in there.