If I know an element and it's inverse, can I learn the modulus? If I know an element $x$ in a group, and it's inverse $x^{-1}$, can I guess the modulo, or with a probability?
As said in the comments above, you have that $x \cdot x^{-1} = 1 \mod n $. 
Another way to say that is $k \times n=(x \cdot x^{-1}) -1$

So no, you can't always find the exact value $n$ from $x$ and $x^{-1}$.

Take $x=12 \ $ and $x^{-1}=17$. You have $x \cdot x^{-1} = 204$. 
So $n$ is a divisor of $204-1=203$, meaning, $ n \in \{29,203\}$ (other divisors are smaller than $\max(x,x^{-1})$ so they can't be $n$)

The only cases where you can find $n$ is when the only divisor of $kn$ greater than $x$ and $x^{-1}$ is $kn$ itself. In this case, $n=kn=(x \cdot x^{-1}) -1 $