The same applies for the JLPT, and not only can you buy past papers (see below), but you can also download them from various websites.
Let's find out where!!!!
Update (May, 2013): We're in the process of adding several more posts to the site regarding JLPT study material. Be sure to check back frequently to see what's new on the front page and what's listed under the LearningJapanese tag.
Because of its 100% Japanese nature, the JLPT is the same for everyone taking it regardless of their native tongue. That's convenient for our purposes, and the main site people seem to go to for past tests is actually in Chinese. Navigating can be a little tricky, but their selection of tests is good. (See below for some other good sites)
The page houses a collection of other papers, plus study resources. It seems like they're trying to charge for newest papers. I can't comment on the legitimacy of the site, so tread carefully if you want to buy.
The resources are of little use in Chinese, but there are some practice questions, too.
In order to simplify matters, here are some direct links to some (free) past test papers. You can try to find more by exploring the site. It's possible the pages might move, in which case you'll have to navigate the page to find their new location.
JLPT level 1 (N1) 2010 July past test
JLPT level 1 (N1) 2009 September past test
JLPT level 1 (N1) 2009 July past test
JLPT level 1 (N1) 2007 past test
JLPT level 1 (N1) 2006 past test
JLPT level 2 (N2) 2010 July past test
JLPT level 2 (N2) 2007 past test
JLPT level 2 (N2) 2006 past test (Nov 2013: Appears to be broken)
JLPT level 2 (N5) 2005 past test
JLPT level 3 (N4) 2005 past test
JLPT level 3 (N4) 2006 past test
JLPT level 3 (N4) 2007 past test
Nov 2013: JLPT level 5 (N5) links from this site have been reported broken or redirected.
There are also answers, and the site is updated quickly enough for you to check your score a couple of days after the test (at the latest).
Check out this site (Nov 2013: Reported broken) which has much better quality scans of all the tests for all levels from 1991-2005. The trouble is, the site is a bit temperamental. If you keep trying every couple of minutes you can download your test, but don't be surprised if you get errors a few times. Weird.
You can also find a large collection of typed-up versions at this site (Nov 2013: Reported Broken).
Bear in mind that these pages can come and go, as you aren't supposed to take the papers out of the test sites: JLPT gets angry. They may force the sites to shut down.
NB: sometimes the scan quality of these tests is not that great. On fairly rare occasions it can be frustrating and almost unreadable.
In such cases you may just want to go ahead and...
Buy published past tests
JAAS sometimes publishes past papers. Here are some links:
Try not to get confused and buy what is known as the JLPT Guide Book. It looks like a test, and acts like one, but is much shorter. It's designed as an entry guide to see if you are ready for typical grade N4 and N5 level questions.
You can buy that here: Amazon America