If you have a basic or a low-paid account on LinkedIn, you can't see the full names of people who are your third level connections. You can't see the names of people outside your network. In LinkedIn Recruiter, which is the solution for corporations, you can see all names.
I didn't realize up till I was at the #sourceIn event yesterday at the LinkedIn HQ in Mountain View, that agencies with the top solution wouldn't see the names either. Now, LinkedIn has decided to open up the names to agencies as well. GREAT! It should be available soon. (They also have some really cool features coming up in the Talent Pipeline.)
However, if you don't have that advanced type of account or don't want to wait until the new feature is implemented for agencies, here is what you can do.
(1) Go to the profile in question.
(3) Look for the message you had just sent, in your "Sent Items" under Inbox.
(4) In your message, see the link to the profile with the name that you couldn't see? Click on it.
There's no need to have a paid account for it to work. It works in exactly 100% of all cases.
(Read more posts on my blog.)
ADDITIONAL INFO. Added on 9/17/2012
Some people can no longer see the "share" link as on the screenshot above. Here is a workaround.
This works just like before. Of course it's a bit more mouse and key clicks but it works.
Also you get to see a link to the person's public profile if it exists and that one will give you even more info in some cases.
Or, just use this link. Here is the exact link that should work (paste the LinkedIn ID where it says so):