The way they didn't realize Bartlet was shot until he started to bleed from the mouth was copied straight from what happened to Reagan.
During the West Wing, Sorkin was a master at dramatizing real life when it worked, or inventing plot points completely when that would play better.
Like Christan Slater destroying his $1400 glass ash tray with a hammer to demonstrate that it's expensive because it's designed to break in to dull pieces and thereby not be dangerous in the heat of naval combat, justifying the cost. Except in real life they'd just use one made of metal.
That so much of it was authentic (to a television standard) gave suspension of disbelief for the made up bits, and seasons 1-4 are consequently some of the best television ever made.
