Possible vs. Optimal
I would agree with those who have stated that heroism is possible in D&D, but I would also maintain that D&D, as published, is not an optimal system for high heroics.
I've come to this view after conducting a two-year experiment with one of my campaigns. The campaign is set in a quasi-Celtic realm and has a decidely heroic thrust. Here's the experiment: to see if the game system used for the campaign has a noteworthy effect on the mood and patterns of play in the campaign.
My group has used four systems for the campaign so far, in the following order:
1. AD&D (2nd ed.)
2. SAGA
3. D&D (3rd ed.)
4. BESM
Throughout the experiment, the campaign's plot has remained consistent, as have the characters, as far as each system would allow.
With each system, and this will come as no surprise to most of you, the mood and patterns of play have changed notably. Sometimes the changes have been subtle, but they've always been noticable.
My group's current view is that SAGA and BESM foster high heroics more than either of the D&D editions we've used, and for that reason, we've decided to finish the campaign as a BESM campaign.
We do think D&D has many strengths, but playing tales of high heroics is not one of them.