in the game "real tennis" the net is parabolic (it appears to hang naturally with loose tension). this was an earlier version of tennis. according to wikipedia it was beginning to evolve as early as the 12th century and has evidence of it being played close to its current form in the 1500s. the modern game didnt start to evolve until the late 1800s. perhaps it came into play as a way of organizing play to be made uniform as it went from something being played for fun to being something played competitively. in making the rule this way, they could be sure that every net (and every statistic and record) was based off the same exact game's rules.
From
http://tennis.about.com/od/history/a/earlyhistory_2.htm:
"The early tennis courts were quite different from the modern "lawn tennis" court most of us are used to. The early game matured into what is now called "real tennis," and England's Hampton Court, built in 1625, is still used today. Only a handful of such courts remain. The net is five feet high on the ends, but three feet in the middle, creating a pronounced droop."
(Wingfield invented the original rules of lawn tennis in 1874 to sell his patented portable tennis game set that included 4 racquets, a net and rubber balls for 5 guineas. One of those hardest hit by the sudden popularity of the new sport was the All-England Croquet Club at Wimbledon. To keep from financial ruin, the club added facilities for [lawn tennis] in 1876.
http://www.trivia-library.com/b/world-history-1874-part-1.htm )
"In 1877, the All England Club held the first Wimbledon tournament, and its tournament committee came up with a rectangular court and a set of rules that are essentially the game we know today. The net was still five feet high at the sides, a carryover from the game's indoor ancestor, and the service boxes were 26 feet deep, but by 1882, the specifications had evolved to their current form."
Just conjecture, but the height of the net and the dimensions of the tennis court evolved as the new rubber tennis ball (made possible by the vulcanization of rubber in 1850), the commercial production of gut strings (sold by Piere Babolat in 1875), and better wood racquets all made the current standard court dimensions and tennis net heights a good test of skills of the best club members of that era.
It is quite remarkable how many of the rules and court/field dimension of our standard sports that evolved so long ago still provide a good backdrop for testing of modern skills.