Is there a possibility that there is no relationship between some inputs and outputs?
Yes and no

The YES

Let's consider at two distributions $F_X(x)$ and $G_Y(y)$ whose joint distribution is simply the product of the the two distributions: $H_{X,Y}(x,y) = F_X(x)G_Y(y)$. This means that the distributions are 100% independent: not just linearly (no correlation), but that there is absolutely no dependence between the two distributions. (Copula, Sklar's theorem, etc.)

The NO

For the above scenario to happen, there has to be absolutely no dependence between the two distributions. At the population level, this is calculated by Hoeffding's test of independence.

$$\theta =\int \bigg{[}H_{X,Y}(x,y) - F_X(x)G_Y(y)\bigg{]}^2 dH_{X,Y}(x,y) = 0$$

For empirical data, this would be tested with the hypothesis test $H_0: \theta=0, H_a: \theta\ne 0$.

Remember how we never accept a null hypothesis of equality because we know that it not quite true?

The YES (again)

It can be the case that, for all practical purposes, the distributions are unrelated. Yes, maybe if a butterfly flaps its wings in Ecuador it will be more likely that there will be an earthquake in Madagascar...but probably not much of an effect. This corresponds to $\theta\ne0$ but $\theta\approx0$ (equal for all practical purposes).