Odds of shock mount failure on vintage Eames lounge  

Well...
I suppose that all depends on what you are going to do with your chair. If you are going to use it and enjoy it, buy some mounts from ebay and get a tube of JBWeld. If you are going to resell it, you might seek professional help. And I wouldn't worry too much about it until they actually break. They can last quite a while with dry mounts.

ZooMob,All of the chairs...
ZooMob,
All of the chairs will experience shock mount failure eventually. It's just a matter of time if you are actually using the chairs. Noone likes to talk about it, but despite looking very cool, most Eames furniture was poorly designed, not very durable and will survive a relatively short amount of time before needing repairs.

Foxxxy, isn't that a bit of an exaggeration?
The glued shock mounts on the chairs fail, but extrapolating to "most Eames furniture" doesn't seem fair. Wouldn't you agree that all the tables, the sofas, the airport chairs, the Aluminum Group chairs, the desk, the storage units, etc., are well-designed and durable?

I agree with fastfwd
I have about 35 eames shells (over 25 years) and not ONE has ever lost a shock mount. Some of those have been "outside" shells too.
And the gazillion or so other Eames designs that I have collected and used, from storage units to tables to other chair designs of all kinds-- all have been spectacularly serviceable. Many were found in "quite used" condition also.
None of my plywood chairs has ever lost a shockmount either.

if you can name a more durable chair...
than the Eames armshell and side shell, please do.
60 years and counting on some of these shells, and all serviceable, and some are still in nearly perfect condition.

Good batches and bad batches
Regarding the fiberglass shells, I largely agree with EamesHead. It generally seems like the epoxy used on the earlier shells was much more durable than later. Only the shock mounts themselves would degrade due to age, exposure to elements, etc., as would be expected with heavy and careless use.
On the later shells (particularly institutional shells used on stacking bases), the shock mounts would frequently pop off, separating cleanly from the epoxy. I believe this is due to the angle of the brackets on the stacking bases, causing significant sideways stress and pressure on the mounts. The frequency of this occurrence could be related to a particular shell manufacturer, epoxy type, or just production run. It would be difficult to pin down the exact source(s) without a large set of data.

but yes...
...the lounger is another story. I bought a pre 1971 lounge chair about 20 years ago, and it lasted for 15 years, and then had the typical mount failure.
I felt lucky to get that much time out of it, considering it had quite a bit of time on it before I owned it.
HB weld is wonderful stuff. I repaired a nelson clock metal arm with it, and it has lasted 20 years so far.

I didn't know that woody...
I have never owned a stacking chair, so that is good to know.
I would imagine that these kinds of issues could run for awhile before they figured out that there was even a problem.
I have had several chairs with crumbling rubber mounts, but even those were always very solidly attached to the shells and were quite functional.

ZooMob, your concern is certainly well-founded.
Though, it can be difficult to assess condition and/or likelihood of imminent failure.
What do you see that makes you say they look "dry"? Are there any visible cracks? Is there evidence of adhesive bond failure? Some rubber-based compounds can remain resilient indefinitely if kept in stable environments, but there are enough horror stories to worry any owner of a vintage chair.
I'd suggest removing the arm pads and have someone sit in the chair and gently flex the back to and fro while watching and feeling for any excessive movement.
It's a judgement call, for sure, and your ultimate decision involves your own sense of what constitutes acceptable risk and personal peace of mind.
FYI, I own a mid-'70's set that had the mounts replaced by HM sometime in the early '90's before I got it and I had one on the seat shell release shortly thereafter. I caught it before it became completely detached. Adhesive bond failure. The shock itself appeared as new, so I re-glued it with PC-7 as per Alfie Hume's recommendation and it has held fast under heavy use (teenagers) for 8 years now. I will admit, however, that I do cringe a bit every time I see one of the cretins flop herself in the chair sideways over the arm.

.
what tktoo said.
And, in case you didn't know this already: If the mount on one of the 'ears' where the back attaches to the seat fails, the back can swing around and snap the other ear in two. Not pretty.

My gosh, our eames lounge chair broke and the bottom panel behind my back has snapped.
Can any one help?, Can i buy a panel from somebody?

Ouch, condolences to you.  
I feel like I'm being reminded / told to have my stupid timing belt changed but I DON'T FEEL LIKE IT, okay?
parallels...
(grumble grumble)