When I was 17 or so I actually tried this as I had read about it and was curious (and willing to experiment on myself ). The results were very surprising. (For reference, 'cold water' meant really cold, likely 10 deg.C or so(*))
After a couple of weeks I could take an ice-cold shower yet feel nothing of the sort. As soon as the water was running, my body started a very vigorous surface circulation and I felt quite fine and not in any discomfort as I did not really feel the cold water; it felt quite invigorating. Obviously this works only up to a point in terms of both temperature and duration.
(*) I found in a regulation that in British Columbia, the cold tap water has to be at most 15 deg.C. However, in an Australian regulation that came up randomly as well, their rule for cold tap water said "to be delivered at a temperature of no more that 40 deg.C'
In any case, the cold showers in my experience are not tied to the temperature of the water that is still comfortable because a different mechanism kicks in.