If you issue them the same award, the award will be displayed for each time you assign it to someone. If you are just wanting to have the "Tourny A" award displayed once for each user, but want it to some how indicate that that person has played in Tourny A ten times, you would have to edit the issue reason for that person. Then it would only show up on their Awards tab (on their profile page).
What I have done on
my site is to have most of my awards (we call them medals) done up in sets of four:
Base Medal
Bronze Medal
Silver Medal
Gold Medal
I can then just replace the medal with the next highest version of it. So, for your Tourny award, you could set the Base award at 1 tournament, Bronze at 5 tournaments, and so on depending on how high you need to go.
Hope this helps some.