APT - Pricing Factors that are not Statistical Factors

Yes at least in theory. Assume there are lot of stocks, each being of type A or type B. Let, the return of type A stock $i$ and B stock $j$ be

$$r_{t,i}^{A}=mu_A+epsilon_t^{i}$$

$$r_{t,j}^{B}=mu_B+epsilon_t^{j}$$

where $epsilon_t^{i}$ and $epsilon_t^{j}$ are independent white noise. Now the return of a stock is uncorrelated with that of any other stock or any factor except its own return. However, being of type A or type B determines the expected return.

You might try to come up with some behavioral reason why e.g. A type stocks have higher expected returns. However, it might be hard to find such characteristics likely because they tend to imply statistical arbitrage opportunities. In practice for example all value stocks tend to move together.