We prove lower bounds on the complexity of maintaining fully dynamic $k$-edge or $k$-vertex connectivity in plane graphs and in $(k-1)$-vertex connected graphs. We show an amortized lower bound of $\Omega (\log n / {k (\log \log n} + \log b))$ per edge insertion, deletion, or query operation in the cell probe model, where $b$ is the word size of the machine and $n$ is the number of vertices in $G$. We also show an amortized lower bound of $\Omega( \log n /(\log \log n + \log b))$ per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.