Actually, there is no reason why you cannot do it!
To confirm this, I just tested it out on a myhosting.com Basic account using only 1 MySQL database, and it works like a charm. When configuring your two blogs, just make sure that you specify a different table prefix for each blog. So for example wp1_ for your first blog and wp2_ for your second. And you can just upload the files to different sub-directories, for example you can have mysite.com/blog1 and mysite.com/blog2.
And there's no reason why you can't have more than 2, as long as you make sure to store the data for each blog in separate MySQL tables.
The only problem with this is that whenever there's a new version, you have to upgrade 2 times
As an alternative, you may be interested in trying something like WordPress MU which allows you to configure multiple blogs with only a single code-base. So it would be similar to what they do with the wordpress.com blog hosting service.