  The cosmological evolution in Nonlinear Born-Infeld(hereafter NLBI) scalar
field theory with negative potentials was investigated. The cosmological
solutions in some important evolutive epoches were obtained. The different
evolutional behaviors between NLBI and linear(canonical) scalar field theory
have been presented. A notable characteristic is that NLBI scalar field behaves
as ordinary matter nearly the singularity while the linear scalar field behaves
as "stiff" matter. We find that in order to accommodate current observational
accelerating expanding universe the value of potential parameters $|m|$ and
$|V_0|$ must have an {\it upper bound}. We compare different cosmological
evolutions for different potential parameters $m, V_0$.
