Basically the title says it all. It would be great if someone could tell give an example using provable security. More information about groups can be found at:
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.
DrLecter gave a good answer, I just wanted to include another well-known example.
The Pohlig-Hellman algorithm can be used to compute discrete logs in groups whose order is a smooth integer. If two parties executing a textbook Diffie-Hellman key exchange use as their modulus a prime $p$ such that $p-1$ has only small factors (is 'smooth') an eavesdropping adversary can use Pohlig-Hellman to recover a secret key and compute the shared secret.
This can be remediated by using a Sophie Germain prime as the modulus - i.e. finding a prime $p$ such that $q = 2p+1$ is also prime. Using $q$ as the modulus ensures that the order of the group is not smooth, thus precluding the use of Pohlig-Hellman.