The ccaption package provides continued captions via \contcaption:

\documentclass{article}
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\usepackage{ccaption}% http://ctan.org/pkg/ccaption
\begin{document}
\section{A section}
\lipsum[1-10]
\begin{figure}[t]
\centering
\fbox{\includegraphics[height=.6\textheight]{tiger}}
\caption{(Continued on the following page.)}% First caption
\label{fig:tiger}
\end{figure}
\begin{figure}[t]
\contcaption{\lipsum[1-2]}% Continued caption
\end{figure}
\lipsum[11-15]
\end{document}
It effective does what you suggest:
\addtocounter{\@captype}{\m@ne}% Add -1 to float counter
\refstepcounter{\@captype}% Step and mark float counter
% ...and the rest
Note though that continuous captions could cause problems when used in conjunction with hyperref.