How could a skilled battle commander like Napoleon Bonaparte lose to Prussians at Waterloo (Belgium) as such?

In the case of Waterloo, Napoleon lost because he met his equals and didn't have enough luck to beat them. Napoleon tried to take down Wellington before Blucher could join him. He failed, but it was (as Wellington afterwards said) "a damned near-run thing." So Wellington stood. In the mean time, Grouchy failed to stop Blucher, arguably because of the "friction" of war rather than an incapacity. But he slowed him down.

But once Napoleon failed to overwhelm Wellington, and Grouchy failed to stop Blucher, the combined forces routed the French. Had Napoleon been a bit better or Wellington a bit worse, Napoleon might have overwhelmed Wellington before Blucher could arrive. Had Grouchy been a bit more aggressive or Blucher a bit less so, Grouchy might have held Blucher for longer and Napoleon might still have won the day.

The other things (Ney charges, sickness, Napoleon's bad decisions on what to attack) are just the normal friction of war. Had Blucher and Wellington lost, we'd be talking about how they should have won were it not for their similar mistakes.

War is not chess and not even the best commanders always make optimal decisions and even when they do, they are not always well executed. And so the battle, which could have gone either way, went to the British and Prussians.

1. We have no evidence that Napoleon was ill during June 1815. None. Nada. Zilch. While on St. Helena, Napoleon wrote extensively about his final campaign, blaming Ney and Grouchy extensively and usually unjustly, demonstrating great ease with finding excuses for the loss. Yet never once does he mention any personal ailments. Neither are there accounts by any other French officers present of any ailments or symptoms being experienced.

   Further, Napoleon was personally and actively supervising the defence of Plancenoit against the attacking Prussian forces throughout the afternoon and evening of that day. It was appropriate for Napoleon to manage this aspect personally, as it would require elements of the Guard infantry that only Napoleon had authority to command. Ney was regarded as a capable commander, and was delegated command of the assault on Wellington's ridge by the Corps of D'Erlon and Reille, that had previously been assigned to him.

2. French and Anglo-Allied forces on the battlefield were closely matched on June 18: 73,000 French with 252 guns against 68,000 Anglo-Allied with 156 guns. The French advantage in guns is actually somewhat greater than the numbers indicate, as almost half the French guns were 12 pounders while the British had only 6 and 9 pounders. However Wellington's ability to deploy most of his forces on the reverse slope of a ridge negated much of the French artillery advantage.

3. Blucher's Prussians would contribute an additional 50,000 men or so over the course of the afternoon and evening, so that by about 7:00 pm the French are outnumbered about 5:3 and only equal in guns.

4. The battle itself, in truth, was lost over the preceding 3 days, the 15th through 17th of June, through sloppy staff work and a touch of treason.

Recall that Napoleon's long time Chief of Staff, Marechal Berthier, had been defenestrated in Switzerland two months earlier. This necessitated finding a replacement, which would be Marechal Soult. Although Soult was more than capable as a commander, he had no experience in a staff role and had not worked closely with Napoleon since at least Friedland 9 years earlier. More staff errors would occur in 3 days by Soult than had occurred in as many years with Berthier:

• "Losing" Lobau's VI Corps on the 16th during Ligny.
• Unclear orders, twice, for D'Erlon during the Battles of Ligny & Quatre Bras and failure to notify Ney that one of his Corps was being re-assigned.
• Unclear orders for Grouchy for pursuit of Blucher following Ligny

And a touch of treason on the morning of the 15th, when the commander of the lead division on Napoleon's right wing defected with his senior subordinates.

Napoleon's plan for the campaign was to drive up the Charleroi-Brussels road between the two Allied armies, and to defeat them both in turn. Two wings of two corps each, under Ney and Grouchy, would engage the forces before them while Napoleon, with the Imperial Guard, cavalry reserve, and Lobau's VI corps would provide the critical mass for victory in turn against both opposing armies.

This was Napoleon's favoured strategy when outnumbered, and had been employed previously with success. It required that at all times the entire French army maintained a central position between the opposing forces. That would fall apart over the ensuing four days.

Bourmont's treason on the night of 14th-15th of June resulted in a delay of several hours for the right wing intended to drive the Prussians east, away from Wellington's forces. This allowed Zieten's Prussian I corps to mobilize forward and oppose the main French advance through Charleroi for several hours on the 15th, instead of immediately having its rear threatened by Grouchy's III and IV French Corps to its east. Consequently the Prussian and Anglo-Allied armies are much closer together on the 16th than might otherwise have been the case.

On the 16th as the battles of Ligny and Quatre Bras are fought, one third of the French force fails to impact either battle as one corps is mis-placed and another counter-marches between he two battles without affecting either. The Prussians are beaten quite severely at Ligny, but the opportunity to completely crush them was availble, and dropped. Further, when the Guard attacks as the mass of decision at Ligny it has the effect of driving the Prussians northwards, towards Wavre. A decisive attack from the west by D'Erlon's corps would have driven them eastwards, back to Namur and several hours further away from Wellington's forces.

Either Lobau's VI Corps should have been deployed forward, on Napoleon's left wing as the mass of decision against the Prussians; or Ney should have been informed that D'Erlon's I Corps was being reaassigned and replaced by Lobau's VI corps. Soult neglected to ensure that all commanders had orders, and failed to inform Ney that one of his corps was being reassigned. A terrible oversight, and unthinkable by Berthier who carried his notebooks obsessively everywhere for just that reason.

Finally, following the victory at Ligny, the orders provided to Grouchy fail to emphasize the necessity of remaining on interior lines vis a vis he Prussian forces. Without that clear guidance, and misled as to the direction of Prussian retreat, he allows the Prussian army to move into a position between him and the main French army.

Then on the morning of June 18th, as Napoleon and his generals enjoy breakfast, the stage has been set for a disaster. Grouchy is only about a 6 hour or so march away - but only by crossing the Dyle River at Wavre where a Prussian Corps is already deployed in opposition. All other routes are hours longer, effectively preventing his command from having any affect on the day's action.

Even still, it would be "the nearest run thing" according to Wellington even though the French would be outnumbered by evening. Yes, there is always friction in war, but there had never before been this much friction with Berthier performing staffwork for an in-person Napoleon. The only remotely close comparison would be April 15 through 18, 1809 - but Napoleon never lost a battle when Davout was present.