Registration and buying the game are two separate things, unless you do it at once.
For registration, it is very fast to complete and all you have to do is fill in some basic info (like email, username, password and etc.)
For buying the game, you need to register first. After that, you have to fill in credit card data or other payment data (like PayPal). I think that it is pretty exciting to buy the game because of the anticipation of playing it.
So the answer to your question is... usually no
(unless there are problems, problems such as the servers to the websites being down).
For me it took apx 24 hours for me to be able to logg in and such.
If you do pay for the game but still can't log in, I would go here to look for help, and you can contact Mojang if you don't find anything, good luck:
http://help.mojang.com/
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Kittens don't grow in minecraft
a few minecraft days
it doesnt
1-2 days
It takes a couple minutes.