You are missing a final closing brace in the headline
definition:
\documentclass{beamer}
\usetheme{CambridgeUS}
\makeatletter
\setbeamertemplate{headline}{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=\paperwidth,ht=2.75ex,dp=1ex,center]{author in head/foot}%
\usebeamerfont{section in head/foot}\insertshorttitle\hspace*{2ex}
\end{beamercolorbox}}% <- this closing brace was missing
}
\setbeamertemplate{footline}{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
\usebeamerfont{title in head/foot}\insertshortauthor
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
\usebeamerfont{title in head/foot}\insertshortinstitute
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
\usebeamerfont{date in head/foot}\insertshortdate{}\hspace*{2em}
\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
\end{beamercolorbox}}%
\vskip0pt%
}
\makeatother
\author{The Author}
\title{The Title}
\institute{The Institute}
\begin{document}
\begin{frame}
test
\end{frame}
\end{document}

I took the liberty to slightly increase the height for the box in the headline (this is just a suggestion, of course, and you can revert to your original settings).
To suppress the headline
and footline
, use the plain
option for frame
; to suppress one template only, redefine it locally to be empty:
\documentclass{beamer}
\usetheme{CambridgeUS}
\makeatletter
\setbeamertemplate{headline}{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=\paperwidth,ht=2.75ex,dp=1ex,center]{author in head/foot}%
\usebeamerfont{section in head/foot}\insertshorttitle\hspace*{2ex}
\end{beamercolorbox}}%
}
\setbeamertemplate{footline}{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
\usebeamerfont{title in head/foot}\insertshortauthor
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
\usebeamerfont{title in head/foot}\insertshortinstitute
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
\usebeamerfont{date in head/foot}\insertshortdate{}\hspace*{2em}
\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
\end{beamercolorbox}}%
\vskip0pt%
}
\makeatother
\author{The Author}
\title{The Title}
\institute{The Institute}
\begin{document}
\begin{frame}[plain]
test
\end{frame}
\begingroup
\setbeamertemplate{headline}{}
\begin{frame}
test
\end{frame}
\endgroup
\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
test
\end{frame}
\endgroup
\begin{frame}
test
\end{frame}
\end{document}

As described in section 3.3 of the beamer documentation, \title
, \author
, etc. accept optional short arguments in square brackets. This short version will be placed into the footline.
\documentclass{beamer}
\usetheme{CambridgeUS}
\usepackage[T1]{fontenc} % european characters
\usepackage{amssymb,amsmath} % use mathematical symbols
\usepackage{palatino} % use palatino as the default font
\title[Polifighe]{Super long title that does not fit in the grey box below}
%\subtitle{An Introduction}
\author[pinco]{\small Author: pinco\\
Assistant: pallo}
\institute[]{\large Polimi\\
\footnotesize Facolt\'a delle merendine industriali\\
\small Corso in cannoli e bomboloni} % COMMAND UNIQUE TO BEAMER
\date{\today}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\end{document}
Note the empty []
after \institute
. Without this, the institute will follow the author in the footline.

Best Answer
You could use the
\institute{...}
field for this purpose: