It is possible, methinks.
1. Hunting is free like in the nordic countries and not restricted to lords and nobility, which means that every peasant knows how to use a bow to get some protein on the table.
2. The crusade of Richard of England retakes Jerusalem. It is soon lost again amidst confusion and bickering among the crusaders, but for a while the English longbowmen, who mowed down the Saracen light cavalry where the heavy French knights had failed, the same Saracen light cavalry that devastated the Kingdom of Jerusalem's army at Horns of Hattin, are the heroes of Europe. Archery is blessed from churches and the image of the brave archer, the commoner who can gain wealth and honour on the battlefield, is fixed in the minds of many a serf and free-holding peasant alike. Archery gains a lot in status, becoming a bit of a fashion among nobles.
3. At Constantinople 1204, the mainly melee crusader army intended to recapture Jerusalem that has been lost after Richard recaptured it face a contignent of Trebizond Archers arrived by ship only a week earlier. The eastern Archer-heavy infantry prevails and the archers are showered in gratitude (and not few offers by beautiful ladies as well as expensive gifts by the citizenry of Constantinople). Through the orthodox world like the catholic before, archery becomes more connected with duty, honour and loyalty, as well as the commoners way of war and road to a better position in society. Serfs are detested, archers are glorified.
4. The Mongols take Buda and Pest and advance onto the Polish plains, eventually turning away from Vienna and Krakow, but leaving eastern Europe understanding the power of the mongolian compund bow, and a few people capable of using it.
5. Under these circumstances, the Swiss and Scot pikemen that emerge during the early 1300s carry BOTH pike and bow. Crecy is seen as a confirmation of what many battles have already told - the bow is superior to the melee cavalry. The success on the battlefield is now dependent on discipline, pike and bow. And when cavalry cannot ride masses of infantry down anymore, and pike against pike becomes an indecisive pokefest, masses of archers are seen as the only way to tear up holes in the pike porcupines. Muskets are still too heavy to be carried together with a pike, a mongolian compound bow and two dozen arrows are not though. The constant threat of arrow rains probably mean that the shield survives, and make the pike against pike even more indecisive.
6. Ultimately, it is not until Gustav II Adolf introduces the light, mobile artillery during the 30 years war that the pike-bow armed Spanish Tertia starts to decline. Artillerists are few, and thus Gustav II Adolf can afford to armour them with finely crafted plate, making them almost immune against the arrows raining down upon them. The artillery tears huge holes in the Tertias, who are unable to react to the small and more mobile Swedish bow-and-rapier infantry that use the holes in the pike porcupines to wreac havoc in the Imperial formations. It is not until the introduction of the flintlock musket with bayonet in the early 1700s that the bow is completely phased out of the European armies, however, ever since the combination of small, tactically mobile melee units and light artillery was introduced, the bow lost its superiority on the European battlefield. It would be seen in the hands of the Cossacks, various militias and assassins in need of a silent weapon for more than 150 years after that though, the most famous last usage probably being by a lone unit of Confederate Long-bow armed infantry at the battle of Shiloh 1862 and perhaps by its last proponent, Nathan Bedford Forrest, in his mounted archers raids in the wilderness of the western theater of the American Civil War.