How to add some negative vertical space before a chapter title?

If you want to re-define the whole layout of chapter headings, you can use titlesec and its \titlespacing command. If you just want to change the vertical spacing, it's easy with xpatch:

\documentclass{report}
\usepackage[showframe]{geometry}
\usepackage{xpatch}%

 \begin{document}


{ %   
\makeatletter
\xpatchcmd{\@makechapterhead}{%
\vspace*{50\p@}}{%
\vspace*{50\p@}
\vspace*{-3cm}}
{}{}
\chapter{A higher chapter}
}%
\noindent

\chapter{A lower chapter}

\end{document}

\chapter issues a \clearpage to start on a new page, and the insertion of a \vspace*, while necessary, inserts content that is sufficient for the \clearpage to be realised. We can avoid this by issuing the \clearpage manually and removing this capability from the regular \chapter command temporarily:

\documentclass{report}

\begin{document}

\chapter{First chapter}

\begingroup
\clearpage% Manually insert \clearpage
\let\clearpage\relax% Remove \clearpage functionality
\vspace*{-2cm}% Insert needed vertical retraction
\chapter{Second chapter}% Regular \chapter
\endgroup

\chapter{Third chapter}

\end{document}

The definition of \clearpage is restored after \endgroup as it adheres to the scoping rules of a redefinition within a group.