Short answer: by the turn of the 20th century, as a result of the increased lethality of arms from the advent of the rifled musket, field artillery, field fortifications, the repeating rifle, and (arguably) smokeless powder.

Here is a great quote from Ivan Stanislavovich Bloch regarding what he perceived as the increasing irrelevance of the bayonet. Bloch was an influential Polish industrialist with the Tsar's ear, and his six volume "The Future of War" describes at length the military tactical and strategic transformations facing the Russian army as of 1899:

To rely simply on the strength of the bayonet in face of modern intensity of fire would be to judge only by the tradition of those times when the bayonet was the last argument in battle. In the Russian army, faith in the bayonet is still sometimes expressed. Among foreign authorities it is no longer met with. The conditions have wholly changed. In former times the result of an infantry battle was thus decided: the combatants advanced upon one another without flinching, exchanged a volley or two, and then rushed upon one another. By such an assault the fate of the battle was quickly decided, the weaker side gave way, and escaped without difficulty if the enemy employed no cavalry. The victors sent two or three volleys after the vanquished, and the battle was over.

The conditions are very different now. Before an attack with the bayonet can be made a zone of murderous fire has first to be passed. Retreat after a repulsed attack upon a fortified position, will be accomplished only after the loss of more than half the attacking force. At such short ranges as will be found in bayonet attacks, almost every rifle bullet will disable one soldier, and often more than one. On a smokeless battlefield the results of such an overthrow will be visible to all. At such close ranges the present covered bullet will penetrate the cranium; but in other parts of the body will have a shattering and tearing effect.

In simpler words, Bloch is reiterating the lethality of arms in his day, and emphasizing the demoralizing effect of the modern battlefield, where soldiers are killed at great distances beyond earshot. Elsewhere in Bloch's work he discusses the horrors of not being able to see one's opponent because of the new invention of smokeless powder, and the horrors of being able to see all the casualties on a battlefield, now that war was no longer fought under the haze of gunpowder smoke.

It is worth mentioning the the U.S. Army, for example, still has a bayonet attachment for the M4 and M16 and sends its soldiers through a bayonet course during Basic Training, a practice that is widely viewed as a waste of time and more for morale-building and fitness than to prepare for bayonet use on the battlefield.

Source: Is War Now Impossible? Being An Abridgement Of The War Of The Future In Its Technical, Economic, And Political Relations (1899) by Ivan Stanislavovich Bloch. Grant Richards: London.