We were waiting for a puppy to come up from our rescue for about 5 weeks which, they told me, was their longest ever 'puppy drought'. Then something happened because our dog's litter came in - 7 pups - very soon followed by several other large litters (including one entire litter of black labs some backstreet breeder couldn't flog off to the unsuspecting).
We were on a waiting list specifically for a puppy - despite having gone there intending to adopt an adult dog (they wouldn't let us as we had an elderly cat).
In the end, we got our pup in 5 weeks, from the first visit, and the rescue were of course, right - a grown up dog would always have been worrying around an old, slow cat unless it had been fostered in a family with a cat.
Puppies come up all the time. Once we were accepted on the list, it was a case of 'first come first served' so we rang up every day to ask if any pups had come in. Eventually, we were told this litter of 8 week old staffy/JRT crosses were in and if we wanted one, to get down there and choose, the next day. (They worm/flea/etc pups before releasing them for adoption, so the pups were there but not viewable for a few days).
We raced there the next morning, and 3 or 4 pups were still available. We had wanted a staffy or staffy cross, so we reserved the one we liked the look of, the most (A very pretty little female, with an eye patch).
Once you are on the list, and approved, you can go for every litter that come up, in theory. It truly wasn't that hard but we feel lucky anyway.