To determine how many non-negative integer values of \( m \) allow the quadratic equation \( x^2 - 2x + m - 5 = 0 \) to have two distinct real solutions, we can analyze the discriminant.

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -2 \)
- \( c = m - 5 \)

The discriminant \( D \) is given by:

\[
D = b^2 - 4ac
\]

Substituting the values, we get:

\[
D = (-2)^2 - 4 \cdot 1 \cdot (m - 5) = 4 - 4(m - 5)
\]

Simplifying this, we find:

\[
D = 4 - 4m + 20 = 24 - 4m
\]

For the quadratic equation to have two distinct real solutions, the discriminant must be positive:

\[
24 - 4m > 0
\]

Solving this inequality:

\[
24 > 4m \\
6 > m
\]

This means \( m \) must be less than 6. Since we are looking for non-negative integer values of \( m \), the possible values of \( m \) are:

\[
0, 1, 2, 3, 4, 5
\]

Counting these values, we have 6 possible values. Therefore, the answer is:

\[
\boxed{6}
\]