Hillary Clinton vs. Barack Obama: Can they both be winning?

It takes delegates to win the nomination but both Hillary and Obama are claiming a lead after Obama's victory in South Carolina. A press release on Hillary's campaign site entitled "Campaign Memo: South Carolina, Florida and February 5" says (emphasis added):

This remains a delegate fight, with 1,681 delegates at stake on February 5th, and 2,025 needed to secure the nomination -- and we are ahead in that fight.

Yet, in Barack Obama's South Carolina victory speech (text as prepared for delivery) he says (emphasis added):

After four great contests in every corner of this country, we have the most votes, the most delegates, and the most diverse coalition of Americans we’ve seen in a long, long time.