I had a lot of 'angst' with my old QS1 getting upgraded... the card was converted to a QS1 from an Orchard/HSBC Platinum MasterCard, and many people told me that because of that it probably would never be upgradable... thankfully that wasn't the case.
I tried repeatedly - and was repeatedly told "there is no offer available". Then finally there was an offer... I could PC to a QS or a Venture One... of course I chose the QS. In my case, the card was nearly nine years old at the time (had been Capital One for three years) so it did take me a while. As it's already been mentioned, some people are able to get a PC in a matter of months... it's very much YMMV.
I'll also add for anybody wanting to get an upgrade to a Signature/World card... usually the easiest way to do this is to PC the card once your credit line is $5k or higher. Several people on here have had success with getting the upgrade this way, since otherwise it tends to be rather elusive. This is actually one of the few times a PC to a Venture One would make sense (then PC back to QS six months later... when you PC back you would keep the Signature/World status).
