This concept exists outside of LinkedIn, but it may be used to great effect when initiating discussions there. Set up a Google alert for whenever that client’s company is mentioned in the press if they are an account you are dying to add to your pipeline. Use the anecdote as a discussion starter if and when it can be related to your line of work.
You may have seen a news release announcing a fresh round of money raised by a company that had previously been unable to investigate purchasing from you due to financial restraints. Contact them, tell them how happy you are for them, and see if they think now is a good time to get back in touch.