Brady has been playing Heineken Cup rugby since the 2004/05 season and, while Ulster have struggled for much of that time, fellow Irish outfits Munster and Leinster have picked up four European titles and appeared in an astonishing 14 quarter-finals and nine semis between them.
And having seen Ulster fail to reach the knockout stages for 11 successive seasons prior to last year's campaign when they lost to Northampton Saints in the quarter-finals, Brady admits it's hard to comprehend just how far they've come in 2012.
"It's incredible. I have been here a long time and for a lot of it Ulster just didn't appear to be making very much progress, if any," the 31-year-old who made his Ravenhill debut a decade ago told the Belfast Telegraph.
"Every year in Europe we were struggling, trying to get out of our group and not making it. Meanwhile Leinster and Munster were doing really well, so we were envious as well as frustrated.
"There just wasn't any sense of direction at all. Everybody wanted to do well, but it just wasn't working out. We were all over the place.
"But now we're getting a turn. We're in a Heineken Cup final and in with a chance of winning it. Bearing in mind where we're coming from and where we were not very long ago, that's incredible.
"When you look at the way things are now, there's just no comparison. I'd say the improvement is down to a combination of factors - recruitment, the culture within the squad, the belief that's there. Things that just weren't there before are in place now."
Ulster's Heineken Cup record since Brady's debut European campaign
2004/05 played 6 won 3 lost 3
2005/06 played 6 won 3 lost 3
2006/07 played 6 won 2 lost 4
2007/08 played 6 won 1 lost 5
2008/09 played 6 won 2 lost 3 drew 1
2009/10 played 6 won 4 lost 2
2010/11 played 7 won 5 lost 2
2011/12 (so far) played 8 won 6 lost 2