MacBook Pro not booting
My question is, if my logic board was truly faulty, would it have been
  possible to boot up when I got it back from the store?


Because, not everything is absolute.

In other words, you may have an intermittent failure of some component.  I't possible it's a failing capacitor, resistor or even IC (Integrated Circuit, aka "chip").   There may be dry (broken or otherwise faulty) solder joints.  You may even have electrical "glue" that has become dry/cracked and is contributing to the intermittent functionality.  

Without putting "hands on" it would be impossible to accurately diagnose and/or repair.  However, IMO, unless you can source a replacement logic board on the cheap, fixing an almost 9 year old machine is going to be cost prohibitive.