Those pesky soft signs mucked me up. I wrote шестдесять лет. The problem I have is that I can't hear them at all.
Don't expect to hear palatalization of t in Russian words for 50 and 60. But ь is always there - think of 5x10 and 6x10 (пятЬ десяТков and шестЬ десяТков, respectively). After all, when we are listening we say 'lissn' rather than "lisTen". It is that kind of thing.
In colloquial speech you'll rather hear even shorter "шисят" meaning 60. The full version (here) sounds like "шиздисят", it's like English "enough"=inaf. However, in many words soft signs can be recognized.