One of the thing that annoys Bush's opponents so much is that Bush does not appear to be the sharpest tool in the shed, and we feel the President should be so. I talked with Bush when he was running, and he wasn't as stupid as he appears on TV under all that scruity (nobody is, everybody reported that Quayle wasn't), but he's not at Clinton's level, for example.
In spite of sharing this feeling, I challenge people to look back in history for the Presidents who were smart at "presidenting." While there have been smart Presidents, the truth is the most admired presidents (in the modern era) seem to be admired not for how clever they were, but for their boldness or the strength of their convictions and leadership.
Intelligence, the voting public seems to feel, is not a top quality for the President, though his advisors should have lots of it. The President's job is to have his vision and decide which of his smart advisors' advice best implements that vision. The vision itself probably can't be all that smart, since the general public has to grasp it and follow it and approve of it.
A stupid President isn't good. He can be fooled by his advisors. He can be fooled or fool himself into thinking that you can invade an occuply a country simply because you have vastly superior miltary force. But smart Presidents can also make stupid decisions on the hard problems. (Americans should know the USA could never have existed if military power were enough to hold a territory.)
We keep an ideal of a philosopher-President like Jefferson, but we haven't had one for a while, we may never have one in the current electoral system. We yearn for President Bartlet perhaps. We like to hear how Clinton was on top of all his advisors told him, but what do you point to and say "This, that was done by Clinton or Carter or Nixon or whoever ... this was really smart."
The Democrats may not understand this. They don't understand how, with Bush shown to be a bit dim every day, he still gets half the country on his side. It's because it's not what that half of the country is looking for.