Matthew,
With one user, the data imports, but with the other it doesn't? Does the other user have all their profile fields filled out in LinkedIn? Is there any extra privacy settings you may have set for that user?
I'm not sure what would be causing the issue you mention, especially since it works for one user. Would you be able to post, or Private Message, the URL of a site where we can register and see what happens?
Also, let us know what version of JLinked and the Social Profiles - JomSocial plugin you are using.
Thanks,
Alex