A wild and desperate attempt to make a play. Sometimes the term carries a hint of showboating.
Babe Ruth (_Babe Ruth's Own Book of Baseball_, 1928) defined "giving it the old college try" as "playing to the grandstand or making strenuous effort to field a ball that obviously cannot be handled."
The term was quickly applied to any effort with limited chances of success.
From _The New Dickson Baseball Dictionary_ (1999) by Paul Dickson.
In a column that appeared in the _Columbus_ (Ohio) _Citizen_ (Nov. 26, 1927) and was quoted in _American Speech_ (Apr. 1930), Billy Evans wrote that "I gave it the old college try" is a term "often used in big league baseball, when some player keeps on going after a fly ball, usually in foul territory, with the odds about ten to one he would never reach it. Teammates of such a player often beat him to it by shouting in unison with the thought of humor uppermost: 'Well, kid, you certainly gave it the old college try,' as he falls short of making the catch. When some player does something that a professional player might not ordinarily attempt, such as colliding with a fielder who had the ball ready to touch him out, in the hope that he might make him drop the ball, regardless of the danger he was courting, someone is sure to say, often ironically, if the speaker happens to be one of the players in the field: 'That's the old college spirit.'"