The billing issue with LinkedIn appears, as of this writing on May 23rd, to have finally been resolved.
Each night for the last several weeks, I have tried without success to update my credit card information and gotten back one of those "Sorry" messages. Finally, when I was about to make yet another attempt, I discovered that the update had actually been done by LinkedIn itself, not by dint of any attempt of my own, but by LinkedIn's own action.
I can settle for that outcome for myself, but I don't know whether other LinkedIn customers might yet be experiencing the problem with getting "Sorry" messages by which I had been beset.
Sadly though, resolution took the exchanging of way too many e-mails and consumed way too much time. Look at this list of back and forth messages which spans twenty-one days:
Still, it looks like I won't be getting booted from LinkedIn after all.