What is even more surprising is how somewhere, someone has a solution to this problem you are facing.
The description can be seen in intellisense suggestions for improved usability of the widgets.
Somehow it just inexplicably stops working the way we expect it to.
Initially I tried Tools - General and checked if ‘Auto List Members’ and ‘Parameter Information’ was enabled (which it was). This caused Intellisense to stop working, but this was just after I had installed Code Rush.
See the article Code Rush Stops Intellisense Working For Visual Studio 2013 for more.
All possible methods with their parameters are now listed in the intellisense.