In the 19th Century, West Point classes were fairly small. There would be 40 or 50 graduates in any given year. Some years, such as 1859, had as few as around 20 graduates.

After graduation, the new officers would be assigned to their first duty stations and respective branches. West Point was primarily an engineering school, so the top cadets were generally assigned to the engineering corps. The, the next group would be assigned to artillery. The third group to cavalry. The last group to Infantry. (Oversimplification. Some officers high in their class would decline their initial possible engineering or staff position for a line position. Others would never leave academic life becoming instructors. So, think of these as broad groupings with a number of personalexceptions each graduating class).

It would be a mistake to think that West Point officers were not sent out to the frontier. The Class of 1870 had 55 graduates and the vast majority spent their entire careers on the frontier. This would still mean that in any given year, there would only be about 30, or so, graduates available spread out 10 cavalry regiments or 25 Infantry regiments that did the bulk of the frontier work.

On paper, an Infantry regiment would consist on 10 companies. Each company would be 97 men and 3 officers. Then, a small command staff. A Cavalry regiment would be 12 companies of 4 officers and 87 men plus a regimental command staff. The regiments were never at their full strength, but even so, there were simply too many officer requirements for just West Point graduates to fill. There were certainly more than that movie would imply. It was not rare. There were graduates in every unit.

Army Lineage Series: Armor-Cavalry Part 1 Army Lineage Series: Part 1 Regular Army

Cullum's Biographical Register of the Officers and Graduates of the United States Military Academy at West Point.