A natural number $n$ is written on the board. Ben plays a game as follows: in every step, he deletes the number written on the board, and writes either the number which is three greater or two less than the number he has deleted. Is it possible that for every value of $n$, at some time, he will get to the number $2020$?
1. **Understanding the operations:**
   - Ben can either add 3 to the current number or subtract 2 from it.
   - We need to determine if, starting from any natural number \( n \), Ben can eventually reach the number 2020.

2. **Analyzing the operations:**
   - If Ben subtracts 2 and then adds 3, the net effect is:
     \[
     n - 2 + 3 = n + 1
     \]
   - If Ben adds 3 and then subtracts 2, the net effect is:
     \[
     n + 3 - 2 = n + 1
     \]
   - If Ben subtracts 2 twice and then adds 3, the net effect is:
     \[
     n - 2 - 2 + 3 = n - 1
     \]

3. **Implications of the operations:**
   - From the above, we see that Ben can effectively increase the number by 1 or decrease the number by 1.
   - This means that from any starting number \( n \), Ben can reach any other number by repeatedly applying these operations.

4. **Reaching 2020:**
   - Since Ben can increase or decrease the number by 1, he can reach any integer, including 2020, from any starting natural number \( n \).

Conclusion:
\[
\boxed{\text{True}}
\]