No the more exercise the better this just means that the puppy is in good health.

 The amount of exercise a puppy needs depends on the individual puppy, the size and breed of puppy.  Extended periods of activity can place stress on the musculo-skeletal system that is developing.  Small, frequent bouts of activity are the best for puppies--they usually like to play hard then sleep hard. 
Five miles does sound long for most puppies.  Also if you have the puppy off leash most dogs actually walk/run farther because they rush ahead then run back to you multiple times.  I do not recommend long hikes or runs with dogs until they are 9-12 months depending upon the dog.  High intensity and high impact exercise can affect development--just like human gymnasts.
Puppies are very sensitive to hypoglycemia and dehydration so bringing along adequate food and water is very important.
There are some developmental diseases that can affect puppies especially large or giant breed dogs--if the lameness persists please have  the pup examined by a veterinarian to assess the joints-especially elbows, hips, and shoulders-as well as the long bones.

 Luvsdog,

Your query reflects that puppy is too young. You can start exercising him when at the age of 5 months. They love to play and exercise on the running guard. Please have a look at the link below.

http://www.runningguard.com/

OR

http://www.dogchannel.com/

Let me know if the given links are fine.



Thanks

 I'm gonna guess its rare but yeah. Does yours have any symptoms?