Let $G$ be a cyclic group of order $n$ and let $\mu = \{x_1,x_2, \dots, x_m\}$ be a sequence of elements of $G$. Let $k$ be the number of distinct values taken by the sequence $\mu$. Let $n\wedge \mu$ be the set of the $n$-subsequence sums.

We show that one of the following conditions holds:

1. $\mu$ has a value repeated $n-k+3$ times

2. $n\wedge \mu$ contains a non-null subgroup

3. $|n\wedge \mu|\geq m-n+k-2.$

We conjecture that the last condition could be improved to $|n\wedge \mu|\geq m-n+k-1$. This conjecture generalizes several known results. We also obtain a generalization of a recent result due to Bollobás and Leader.