Sure, I can help you write a test case generator for the given problem statement. Here's a Python function that generates test cases for this problem:

```python
import random
import string

def tcgen():
    # Generate N, the number of key presses
    N = random.randint(1, 5000)
    
    # Generate s, the resulting string after N key presses
    # Ensure that the length of s is between 1 and N
    s_length = random.randint(1, N)
    s = ''.join(random.choice('01') for _ in range(s_length))
    
    return N, s

# Example usage:
N, s = tcgen()
print(N)
print(s)
```

This function generates a random integer \(N\) within the specified range and a string \(s\) consisting of '0's and '1's, ensuring the length of \(s\) is between 1 and \(N\).

**Explanation:**

1. **N Generation**: The number of key presses \(N\) is randomly selected between 1 and 5000.

2. **String \(s\) Generation**: 
    - The length of \(s\) is chosen randomly between 1 and \(N\).
    - The string \(s\) is generated by randomly choosing '0' or '1' for each character in the string.

This function ensures that the generated test case adheres to the problem constraints. You can use this generator to create multiple test cases and validate your solution against them.