They happen, and the FDA has placed a big warning label
on one commonly prescribed for cats, called Metacam.

The point here is that reactions happen fairly often- and you as
a concerned and involved pet owner need to be aware of them.

On the FDA site there is some specific info:

What Veterinarians Should Tell Clients About Pain Control and Their Pets

by Michele Sharkey, DVM, Office of New Animal Drug Evaluation;
Margarita Brown, DVM, Office of Surveillance and Compliance; and
Linda Wilmot, DVM, Office of New Animal Drug Evaluation
FDA Veterinarian Newsletter 2006 Volume XXI, No I

Non-steroidal anti-inflammatory drugs (NSAID) are commonly
prescribed and extremely effective pain control drugs for pets.
Like most drugs, they do cause side effects, some serious.
Veterinarians are in the best position to inform their clients
about these side effects, so the clients can take better care
of their pets. And, pet owners expect veterinarians to explain
all potential risks of medications.

In the United States, NSAIDs commonly used in dogs include
ETOGESIC (etodolac), RIMADYL (carprofen), METACAM (meloxicam),
ZUBRIN (tepoxalin), DERAMAXX (deracoxib), PREVICOX (firocoxib),
and NOVOX (generic carprofen). These drugs have been approved
by the Food and Drug Administration’s (FDA) Center for Veterinary
Medicine (CVM) for use in dogs. (You can get more information
about these drugs by going to CVM’s website,
Once there, go to the “Green Book” navigational button, where
you can look up the drugs by their brand names or active ingredient.)

Most adverse reactions are mild, but some result in permanent
impairment or even death. If the client can recognize a possible
reaction and stop the medication while seeking veterinary attention for the dog, the client may make the difference between a good outcome and a disaster.

The most common side effects from NSAIDs include vomiting, loss
of appetite, depression/lethargy, and diarrhea. Some side effects
can be serious, especially if the drug is not used according to
labeled directions, resulting in the need for medical care.
Serious adverse reactions include gastric ulcers, kidney and
liver problems. Death may result in some instances.

An informed dog owner is the best defense against serious side
effects from NSAIDs. The veterinarian is the most qualified
source for information regarding NSAID use and a dog’s care.
Owners should not hesitate to ask questions and inquire about
possible side effects or signs to watch for when treating a
dog. A Client Information Sheet, which a veterinarian should
give the pet owner whenever an NSAID is prescribed, serves as
a reminder of this information for use at home.