Using ArcHydro, I set stream definition at 5000, 20x20 cells (approx. 50 acres). I placed points at origins of each subsequent stream with batch point generation tool. Then used batch watershed generation, which gave me 23 watersheds ranging in size from approx. 50 acres to 82 acres. Why the range in watershed sizes?
The key here is that flow may proceed into your origin point from multiple directions (depending on your flow direction method). Considering a D8 method, the could be as few as one inflow cell or as many as 7 (and one outflow). In the case of one inflow cell, all 50 acres upstream accumulate to this one cell and therefore initiate a stream, by your definition. However, multiple other flow paths could flow into this same cell, but accumulate less than 50 acres upstream. Therefore, the cell in which your origin point has been placed has accumulated the requisite 50 acres from one upslope cell as well as additional area from other inflowing cells.