Greetings,
If you are using phpmyadmin you are limited by HTML for display and PHP for dealing with the quotes when the come back out of your database. My thoughts are that the problem is somewhere between PHP and HTML.
Looking at the page source to see if the value is the same as what is being displayed would be a good first step (is it truncated in the source as well?), but I would suggested using the tools that come with mysql (Im talking command line) to validate that the double quotes are truncating the values being inserted.
Normally if it is and issue with quotes being inserted, be them single or double, then DBI or the corresponding DBD will warn you about that.
Just a thought.

-InjunJoel