Short, but seems an appropriate response given the lack of explanation in the linked article.

IEEE 754 double-precision (64-bit) can only represent even numbers in the range 2^53 to 2^54.

Haskell lets us look inside them fairly easily:

  > exponent 9999999999999999.0
  54
  > significand 9999999999999999.0
  0.5551115123125783
  > exponent 9999999999999998.0
  54
  > significand 9999999999999998.0
  0.5551115123125782
  > 2^54 * 0.5551115123125783
  1.0e16
  > 2^54 * 0.5551115123125782
  9.999999999999998e15