You can use the fitting library of the tcolorbox package. It provides a macro called \tcboxfit which fits the font size of the content to the dimensions of the box. Alternatively, the option fit for tcolorbox does the same trick.
Note that the used font has to be resizable to all dimensions (vector font) to achieve the correct resizing.
The first example shows a framed box (can be adapted in many ways), the second one a box without a frame (the red border is just for displaying the size), the third example displays a box with title and fixed height.
\documentclass[a4paper]{article}
\usepackage[many]{tcolorbox}
\usepackage{lmodern}% or any other vector / postscript font
\usepackage{lipsum}
\begin{document}
\tcboxfit[width=6cm,height=6cm,nobeforeafter,
before=\noindent]{\lipsum[1]}
%
%
\begin{tcolorbox}[fit,width=6cm,height=6cm,blank,
borderline={0.4pt}{0pt}{red!20!white},
watermark text={6cm $\times$ 6cm},nobeforeafter]
\lipsum[1]
\end{tcolorbox}
\bigskip
\tcboxfit[height=6cm,title={This box has a height of 6cm},
colback=blue!5!white,colframe=blue!30!black,
before=\noindent]{\lipsum[1]}
\end{document}

\parboxwhen the text overflows? You should get anoverfull hboxmessage in that case. – Qrrbrbirlbel Nov 10 '13 at 22:52\parbox[t][2cm][t]{2cm}{\lipsum[1]}- and there is no warning here. – masu Nov 11 '13 at 08:48