No Line Break in Author Command? It goes off the page!

Add \\ at suitable spots: here one suffices after the fifth author.

\documentclass[conference]{IEEEtran}

\begin{document}
\title{An Important Conference Contribution}

\author{\IEEEauthorblockN{Author One\IEEEauthorrefmark{1},
Author Two\IEEEauthorrefmark{2}, Author Three\IEEEauthorrefmark{1},
Author Four\IEEEauthorrefmark{2}, Author Five\IEEEauthorrefmark{1},\\
Author Six\IEEEauthorrefmark{2}, Author Seven\IEEEauthorrefmark{1},
Author Eight\IEEEauthorrefmark{2}, Author Nine\IEEEauthorrefmark{3} and
Author Ten\IEEEauthorrefmark{4}}
\IEEEauthorblockA{Department of Whatever,
Whichever University\\
Wherever\\
Email: \IEEEauthorrefmark{1}[email protected],
\IEEEauthorrefmark{2}[email protected],
\IEEEauthorrefmark{3}[email protected],
\IEEEauthorrefmark{4}[email protected]}}
\maketitle

\end{document}

Use a \parbox of \linewidth and use % to hide spurious spaces.

\documentclass[conference]{IEEEtran}

\begin{document}
\title{An Important Conference Contribution}

\author{%
  \IEEEauthorblockN{%
    \parbox{\linewidth}{\centering
      Author One\IEEEauthorrefmark{1},
      Author Two\IEEEauthorrefmark{2},
      Author Three\IEEEauthorrefmark{1},
      Author Four\IEEEauthorrefmark{2},
      Author Five\IEEEauthorrefmark{1},
      Author Six\IEEEauthorrefmark{2},
      Author Seven\IEEEauthorrefmark{1},
      Author Eight\IEEEauthorrefmark{2},
      Author Nine\IEEEauthorrefmark{3} and
      Author Ten\IEEEauthorrefmark{4}%
    }%
  }%
  \IEEEauthorblockA{%
    Department of Whatever, Whichever University\\
    Wherever\\
    Email: \IEEEauthorrefmark{1}[email protected],
    \IEEEauthorrefmark{2}[email protected],
    \IEEEauthorrefmark{3}[email protected],
    \IEEEauthorrefmark{4}[email protected]%
  }%
}
\maketitle

\end{document}