To prove that the expression

\[
M = \frac{x}{x+y+z} + \frac{y}{x+y+t} + \frac{z}{y+z+t} + \frac{t}{x+z+t}
\]

is not a natural number for \( x, y, z, t \in \mathbb{N}^* \), we can analyze the components of \( M \).

1. **Understanding the Terms**: Each term in \( M \) is a fraction where the numerator is a natural number and the denominator is a sum of natural numbers. The denominators are strictly greater than the numerators since they include additional positive terms.

2. **Range of Each Fraction**: Each fraction \( \frac{x}{x+y+z} \), \( \frac{y}{x+y+t} \), \( \frac{z}{y+z+t} \), and \( \frac{t}{x+z+t} \) is less than 1 because the numerator is less than the denominator.

3. **Sum of Fractions**: Since each term is less than 1, the sum \( M \) will be less than 4 (the sum of four fractions each less than 1).

4. **Non-integer Result**: The sum of fractions that are all less than 1 cannot equal a natural number. The only natural numbers are 1, 2, 3, etc., and since \( M < 4 \), it cannot equal any of these.

5. **Conclusion**: Therefore, \( M \) cannot be a natural number, proving that the expression does not yield a natural number for any positive integers \( x, y, z, t \).

Thus, we conclude that \( M \) has values that are not natural numbers.