When referring to Gettysburg, the big mistake is well known: Pickett's Charge. I wouldn't call Pickett's charge reasonable. Could you make that argument for July 1st and 2nd, yes. But he knew what awaited him across that mile of open ground on July 3rd. I don't have to get into details because we all know what happened, but that charge is constantly debated because of the loss and the circumstances behind the man who ordered it.
Oh, I can absolutely defend it in terms of what Lee knew at the time.
So on the first day, Lee had mauled two corps of the Army of the Potomac (thanks to the circumstances of the encounter battle). This is something that is known as far as Lee is concerned - he's badly damaged the numerical superiority the AotP has over him, and in some respects it is now an even battle.
On the second day, Lee attempts an enveloping attack which turns into an attack en echelon, which in turn collapses because two division commanders go down. This is good Napoleonic set-piece work - the hard part of the process (the first bit of the echelon attack) works because Longstreet does great in his sector and draws in all the Union reserves, but the easy part (the bit where the angle around Cemetery Hill is hit by a converging attack at 2:1 odds, and it happens to be about to fall on the weak 11th Corps) doesn't go through because of an error.
Regardless, during this battle the data Lee has seems to suggest that the Army of the Potomac is in a weakened state. On both flanks he does good damage, more damage than he takes despite attacking, and in the 11th Corps sector he drives back Union brigades while along the line of Cemetery Ridge one of his brigades gets quite far before being repulsed.
So on the third day Lee has the opportunity to try and finish the job (i.e. win the battle) or go home without using his reserves. If there is even a 20% chance of finishing the job then it is a good idea to try, because the Army of the Potomac is in many important respects weaker now than it will ever be again.*
Lee has already tried an echelon attack, but there are other tricks in the Napoleonic playbook - the manoeuvre sur les derrieres and the breakthrough of the centre. The manoeuvre sur les derrieres (effectively a flanking manoeuvre) is not really an option because Lee's army is already quite widely stretched, but he has unfought reserves in the form of Pickett's troops, newly arrived to the battlefield.
What Lee does on the third day is therefore (and quite clearly) an attempt at a Napoleonic push through the centre. He engages Meade on both flanks (and Meade duly engages 6th Corps, breaking up his reserves into small penny-packets which are no longer able to oppose Lee's movement). Then Lee mounts a grand-battery bombardment before launching a strong force in two waves to reach and break through the enemy centre.
Yes, that last sentence isn't what actually
happened, but it's what Lee's
orders were. In concept this is good Napoleonic stuff, if risky, but - well, if there was a 20% chance of victory it was worth it, and with both waves going in to make the attack stronger in mass I think it might well have that high a chance. The fact that it is in keeping with good Napoleonic battle management (of the sort which doesn't apply on every battle in the Civil War!) and that Meade made pretty much the wrong move in terms of countering it means that it was a sound option to try - even if it didn't work.
* routed corps, low numbers, the whole army hasn't won a victory since last September, and it's only got two lines of retreat where one is insufficient to handle the entire army; if Lee can force the Army of the Potomac to retreat then it's in a bad shape.