What does it mean when proof by contradiction doesn't lead to a contradiction?

Your goal is to show that $p$ is false.

If $p \implies q$ and $q$ is true.

We can't conclude if $p$ is true or false. Hence, we get an inconclusive situation.

It means that your precise approach does not work, but since this does not provide a counterexample it means the question would still be open.

Seeing $170-165$ is so small, there may be a way to save your proof. Try $$a_i + a_{i+1} + a_{i+2} \le 16$$

leading to $$3 \cdot (a_1 + a_2 + \dots + a_{10}) \le 16 \cdot10$$ and $$165 \le 160$$ for a contradiction

A proof by contradiction functions by saying "if A is false, B must be true. I can prove that B is false, so A cannot be false."

You proved that B is true. This means that A could be false, but is not necessarily so because we have made no statements that relate the truth of A to the truth of B.