Lumineers over Existing Veneers?

Congratulations! 28 years for a set of veneers is EXCELLENT! Bonding has significantly improved over 3 decades of dentistry.  Bonding OVER your existing veneers is not very predictable, as you would have 2 different bonded layers... Not a great go-to solution. Removal of existing veneers, with a new set fabricated, is the most predictable solution. Ask your dentist for REAL patient examples, and what laboratory he uses!As with all information, ask a dental professional prior to proceeding with any treatment.

No, placing porcelain veneers over existing veneers is not a viable option.  Just have the old veneers removed and new veneers placed.  If you want thin veneers there are several options, Lumineers is only just one of them.  Many different labs offer minimal thickness veneers.
The veneers are removed and the teeth prepared quite easily.  Local anesthetic is given, the veneers are easily drilled off, the dentist can easily see where the old veneers ended and the underlying tooth under it begins.  The dentist will extend the tooth preparation to cover the areas of your roots that are now exposed because of the recession you experienced over the years.

By the way, I would discuss EMax veneers with your dentist, as they can be made almost as thin as Lumineers, but made of an advanced porcelain material that is much stronger, and thus most cosmetic dentists expect them to last many times longer. 

Funny that you said 28 years ago, as 1985 was the year that porcelain veneers were first introduced to dentistry.  Is there any possibility that you did not get porcelain veneers, but rather direct composite (plastic) veneers?

Do not place veneers over existing veneers.  Have the old ones removed and new ones placed.  A good cosmetic dentist has done this many times and can give you a great result.