Because it seems people just keep adopting a bunch of mine and while I've only used up one mb of my database out of 100, it's only been 3 days and I'd hate to run out of room in the future.
Yes, you can by hard-coding this value into the script. Please see this post which describes how to limit the number of adoptions per user to 1 per user.
To set the number, in the provided code find:
PHP:
if($num > 0){
Replace 0 with the number for the maximum adoptables a user can have.
Also, if database space is filling up fast you can clear the contents of your adopts_vote_voters table. Be sure to clear it late in the day though as it keeps track of who voted to level up an adoptable and if you clear it mid-day some people can vote for a pet twice in a day, but the data from old dates is not needed anymore.
You can clear by running this query in PHPMyAdmin: