"They ran into Giguere" is a good answer, but I think there's more to it.
Keep in mind that Scotty Bowman retired after 2002 (when the Red Wings won the Cup). The 2003 edition seemed far more focused on the regular season than they should have been. (This is pure speculation, but maybe new coach Dave Lewis wanted to show that it wasn't just Bowman who could put together a 100+ point team?)
As of February 9th, the Red Wings had a 27-17-9-2 record. A bit below their potential, but they weren't in a bad spot. They were still T-7th in leaguewide standings (and #4 in their conference). They could have cruised into a playoff spot. Instead, they played their best hockey of the season during the final months. From Feb 10th onwards, they had the best record in the league (21-3-1-2). I think Lewis should have had the team take their foot off the gas pedal. Why do you need the reigning Stanley Cup champions, who had one of the oldest cores in NHL history, to win 83% of their games down the stretch?
Look at how hard Lewis rode Lidstrom and Chelios during the final two months. They ranked 3rd and 5th in the entire league in SH TOI during that period (Lidstrom was 5th in total TOI; Chelios was lower because he didn't go on the powerplay, but he was near the top in terms of harder ES+PK minutes). They would have been better rested for the playoffs, had they got a minute or two less per game. Why did Fedorov get 24:26 in the 2nd last game of the season, a completely meaningless match against a non playoff team? In the final game of the season (against a non playoff team, and with the division title already locked up) why did Lidstrom get 29:23 (especially when he was north of 31 minutes the game before!)
There are other examples, but the point is - I think Lewis lost perspective on what was important for this team, and rode his stars too hard in the regular season, which would have caused problems in the playoffs, Giguere or not.