Can anyone give an example where (asymmetric) crypto can go wrong due to selection of wrong groups?
Basically, every time you choose a group where the required hard problem is not hard, then you will run into a problem. Even if we have a problem instance that is of size that is considered secure in the setting of asymmetric cryptography. 

Lets for instance implement a discrete logarithm style cryptosystem in the group $Z_n$ with addition and let $g$ be a generator. Then the discrete logarithm problem in this group is given $g$, $y$ you have to find the discrete logarithm $x=\log_g(y)$, i.e., compute $x$ such that $y\equiv xg \pmod n$. However, you can efficiently solve this linear congruence by using Euclid. 

Consequently, the discrete logarithm problem in the group $(Z_n,+)$ is easy and thus every discrete logarithm based cryptosystem will be insecure in this setting.