To solve this problem, we need to determine the maximum number of bishops that can be placed on an $8 \times 8$ chessboard such that each bishop is threatened by at most one other bishop. The key here is to understand the attacking capability of bishops on a chessboard.

Bishops move diagonally, which means they control all the squares on the diagonals they occupy. Therefore, if a bishop is placed on a particular square, it can threaten any other bishop on the same diagonal.

### Step-by-step Explanation:
1. **Diagonals on an $8 \times 8$ board:**

- There are $15$ possible diagonals in each diagonal direction on an $8 \times 8$ board (from the top-left to bottom-right, and from top-right to bottom-left).

2. Placing Bishops:

- We need to place the bishops such that no more than two bishops are on the same diagonal and every bishop is only threatened by at most one other bishop. That means at most two bishops can be placed on each diagonal without violating the threat constraint.

3. Strategy for Maximum Placement:

- The objective is to utilize the longer diagonals more effectively because these allow placing two bishops threatening each other.

- Specifically, for each longest diagonal, we place two bishops as they fully utilize the diagonal's length.

4. Calculation for Maximum Bishops:

- The board can be broken into two sets of diagonals—one set increasing from bottom-left to top-right and another set decreasing from top-left to bottom-right. Both sets have $15$ diagonals each.

- Since no two bishops should be on the same cell, and adding more than two bishops on any diagonal would cause more threats than allowed, we can place two bishops on several diagonals and strategically use shorter diagonals as needed.

5. Final Determination:

- To maximize, we primarily use $10$ diagonals that can accommodate two bishops each. This accounts for $20$ bishops:

$$2 \times 10 = 20$$

- Thus, it can be shown through examples and layouts that the maximum number of bishops we can place under the given conditions is indeed:
$$\boxed{20}$$
6. Conclusion:

- The placement ensures that at most one other bishop threatens any placed bishop, utilizing diagonal symmetry and bishop placement strategies effectively for maximized output.