In WP3.6, when using the Settings API for a theme, I cannot save options that are text.
When bypassing the Settings API (do it old school, very old school), the settings are saved.
I noticed this while developing a theme. So I took a theme form Chip Bennet, that should at least work normally, and there it also fails. All other types of options, checkboxes, radio-buttons, are saved. The text ones not.
Before I report a bug, and trust me reporting a bug on something that is not happening, and does not throw any php, or other error is troublesome, I wonder if anyone else has this problem.
To make it comparable, I tested it with the Theme Oenology, by Chip Bennet, I tried to enter my twitter account, and save the settings. The settings are saved, except for the twitter account (the only text option that makes sense for me, but the other text options weren't saved either).
I tried on a multisite install, on a single site install, and on a fresh install. All with the same result.