I used \stackinset AND routines from the scalerel package. What it insets is a \bigcap that has been scaled down to 1.6ex in height (which narrows it to the proper width), and then vertically stretched to 2ex in height. The manipulated inset is placed, horizontally centered, .15pt above the base of the unaltered \bigcap.
\documentclass{article}
\usepackage{amsmath}
\usepackage{scalerel}
\usepackage{stackengine}[2013-10-15]
\newcommand\bigdoublecap{%
\mathop{\ensurestackMath{\stackinset{c}{}{b}{.15pt}%
{\stretchto{\scaleto{\bigcap}{1.6ex}}{2ex}}{\bigcap}}}
}
\begin{document}
\[
\bigdoublecap_{i=1}^n A_i
\]\end{document}

\capwould make it better to read. – C-Otto Oct 17 '13 at 22:54