I am having the same problem and getting the same error messages, at the site I administer.
The messages are "Saving failed: AJAX call successful, but unclear data. Try again while holding down the “Shift” key." - even when you follow these instructions, the next message "Error: The table could not be saved." comes up.
I am running TablePress 1.1.1 on a WP 3.7 installation. The tables are not very large (2 or 3 columns, and say up to 12 rows) and when you try to save changes to any of those tables, it appears some of the changes sometimes then appear on certain browsers like Chrome, even if the error messages come up.
Other times, the changes really are not saved.
I have a lot of plugins on the site so it is difficult to identify any which may be causing a problem (unless you know of any known issues):
Dagon Design Sitemap Generator
Dynamic Headers by Nicasio Design
Enhanced Meta Widget
Eventbrite for The Events Calendar
Facebook Social Plugins
Fast Secure Contact Form
Get Picasa Albums
Gwyn's Imagemap Selector
Jetpack by WordPress.com
The Events Calendar
Toppa Plugins Libraries for WordPress
WP Facebook Open Graph protocol
WP Render Blogroll Links
I was about to donate to your plugin having tested it out a bit, but wanted to confirm this issue has been fixed before I do that.