In this paper, we disprove the long-standing conjecture that any complete geometric graph on $2n$ vertices can be partitioned into $n$ plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which \emph{cannot} be partitioned into plane spanning trees (or even into arbitrary plane \emph{subgraphs}), including a complete description of all possible partitions (into plane spanning trees). Furthermore, we show a sufficient condition for \emph{generalized wheels} to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars.