What is a Fifty Mission Cap? The question "What is a Fifty Mission Cap?" has been asked several times.
Listed below are explanations provided on various USENET newsgroups:

From doug_blair@mitel.com Sat Mar 29 09:58:14 EST 1997
Article: 33271 of alt.sports.hockey.nhl.tor-mapleleafs
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From: Doug Blair 
Newsgroups: alt.sports.hockey.nhl.tor-mapleleafs,alt.music.tragically-hip
Subject: Re: Fifty Mission Cap
Date: 20 Mar 1997 15:37:38 GMT
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Stephen L. Legge wrote:
> 
> In article <33306AB7.55B5@ns.sng3ympatico.ca>, Clements
>  wrote:
> >What is a "Fifty Mission Cap"? Is there any connection between the "Fifty
> >Mission Cap" and Bill Barilko or is just a coincidence that Gord Downie
> >keeps his Bill Barilko card under his "Fifty Mission Cap"?
> >
> >CKC
> 
> I may be *completely* wrong, but I always assumed it is something an air force
> pilot receives after completing 50 sucessful missions.
> 
> Perhaps it's a Downey family heirloom?  Just a wild guess.
> 
> Stephen LEAFS Legge
> SLEGGE@cs.mun.ca - www.cs.mun.ca/~slegge
> St. John's, Newfoundland
> 
> "I would rather have played for the Leafs than
>  be prime minister of Canada." -- former PM Lester Pearson

You are entirely correct.  A fifty mission cap is what WWII ace's wore
to identify themselves as pilots with over 50 missions flown.  They were
"allowed" to beat the crap out of their otherwise militarily proper
flight caps.  This obviously made them more identifiable and therefore
respected by the rookies.  Other than making for an interesting song
there is no connection between Bill Barilko's hockey card and 50 mission
caps.


From thillt@thcsf.com Sat Mar 29 09:58:28 EST 1997
Article: 33283 of alt.sports.hockey.nhl.tor-mapleleafs
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Date: Thu, 20 Mar 1997 10:42:06 -0500
From: Tom 
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Clements wrote:
> 
> What is a "Fifty Mission Cap"? Is there any connection between the "Fifty
> Mission Cap" and Bill Barilko or is just a coincidence that Gord Downie
> keeps his Bill Barilko card under his "Fifty Mission Cap"?
> 
> CKC
Gord Downie likes to mix things up in his songs.  The song is about both
Bill Barilko and 50 Mission Cap which is from the old fighter pilots
who's leather hats were worn in after each mission.  The Fifty Mission
Cap is a really good worked in cap.  "I worked it in to look like that"
could be taken as someone wanting their cap to look like it had been on
fifty missions.  I heard him say he didn't really know the history of
Barilko scoring and disappearing so when he read the card it inspired
him on that front and I believe that that ( don't quote me ) but the
Fifty Mission Cap idea came from the fact that planes were used to
search for Barilko and that is where that tangent came from.  A lot of
the Hip's songs go similarily with a few meanings and interpretations. 
Sorry for rambling just my two cents.
Back to the Leafs.  I thought they looked good until the Lindros show
took over last night.  We need some big forwards.  The Legion made us
look like Dwarves out there last night.
Cheers.
Tom

Fifty Mission Cap lyrics on The Tragically Hip web site.