How many players or draft choices were involved in the largest trade
in NFL history?  Is the answer 16, 18, 19, or 15?

The answer is 18.

"1989 
Jimmy Johnson, Landry's replacement in Dallas, proceeds to makes the
largest trade in NFL history when he sends RB Herschel Walker to
Minnesota. The total deal involves 18 players and draft choices and
also the San Diego Chargers."

American Football Timeline
http://www.geocities.com/Colosseum/6513/football/timeline.html

"The largest trade in NFL history, when the number of players and
draft picks were combined, was the Dallas Cowboys' trade of Herschel
Walker to the Minnesota Vikings in 1989 -- which involved 18 players
and draft choices."

Cached copy, "Cowboys Fun Facts"
http://64.233.167.104/search?q=cache:KB4e5RenMPgJ:theboys.com

Google search strategy:

Google Web Search: "largest trade in nfl history"
://www.google.com/search?hl=en&q=%22largest+trade+in+nfl+history

I hope this is helpful. If anything is unclear or incomplete, please
request clarification; I'll gladly offer further assistance before you
rate my answer.

Best regards,
pinkfreud