Toby,
Are you sure that the user's have selected the box "Allow profile import" or whatever it's called on the registration page? If they don't select that, the only information that we're allowed to fetch from LinkedIn is their full name. Currently, we don't record if the user selected that option, so unfortunately, it may be a little difficult to debug. That's my best guess though.
Can you let me know if for the users that 'something' is missing, is it all missing.. or just pieces? If it's everything, the checkbox for profile import would explain it. If it's just bits and pieces, then I'm not sure and we'd have to investigate further for a cause.
Thanks,
Alex