Can plastic leak into water from a HDPE "2" plastic container at room temperature? Here in USA supermarkets often sell water in plastic 1 gallon jugs. These jugs seem to all be stamped with the text "HDPE" (meaning that the material is High-density polyethylene) and a resin number of "2" on the bottom.

I had one of these jugs, containing distilled water, on a counter in the kitchen, and another one in the refrigerator. For some reason the water on the counter had a light plastic taste, whilst the water in the fridge did not.

Could this taste be because of plastic leaking into the water? Or what else could cause the plastic taste?
HDPE 
has a molecular weight more than several hundreds of thousands, 
"leaking" is plainly impossible. 
This taste is the tase of stale water, nobody knows 
what that taste is actually. 
Maybe its just the taste of not cold enough water,
lacking some dissolved air, or some bacteriae 
develop after contact with air and light. 
(google for pseudomonas) 
The only substance which ever got into the jug prior 
to the water is the air (or some other gas?) used to
inflate the jug in production. 
But that cause would impair the taste of the water in 
the fridge as well.

BTW why do You drink such botteled water? 
What about the tap water in your town/city/village?