You would think that predicting a binary response variable perfectly would be a "good thing," but it can create problems in estimating logistic models as these examples demonstrate.
Examples
use http://www.gseis.ucla.edu/courses/data/honors, clear
logit honors female, nolog
Logit estimates Number of obs = 200
LR chi2(1) = 3.94
Prob > chi2 = 0.0473
Log likelihood = -113.6769 Pseudo R2 = 0.0170
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | .6513707 .3336752 1.95 0.051 -.0026207 1.305362
_cons | -1.400088 .2631619 -5.32 0.000 -1.915876 -.8842998
------------------------------------------------------------------------------
logit honors female lang, nolog
Logit estimates Number of obs = 200
LR chi2(2) = 60.40
Prob > chi2 = 0.0000
Log likelihood = -85.44372 Pseudo R2 = 0.2612
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 1.120926 .4081028 2.75 0.006 .321059 1.920793
lang | .1443657 .0233337 6.19 0.000 .0986325 .1900989
_cons | -9.603365 1.426404 -6.73 0.000 -12.39906 -6.807665
------------------------------------------------------------------------------
generate h2 = honors
replace h2 = 0 if ~female
logit h2 female
note: female != 1 predicts failure perfectly
female dropped and 91 obs not used
Logit estimates Number of obs = 109
LR chi2(0) = 0.00
Prob > chi2 = .
Log likelihood = -68.41892 Pseudo R2 = 0.0000
------------------------------------------------------------------------------
h2 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | -.748717 .2051461 -3.65 0.000 -1.150796 -.346638
------------------------------------------------------------------------------
logit h2 female lang, nolog
note: female != 1 predicts failure perfectly
female dropped and 91 obs not used
Logit estimates Number of obs = 109
LR chi2(1) = 40.59
Prob > chi2 = 0.0000
Log likelihood = -48.121483 Pseudo R2 = 0.2967
------------------------------------------------------------------------------
h2 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lang | .1625339 .0319694 5.08 0.000 .099875 .2251928
_cons | -9.476008 1.768424 -5.36 0.000 -12.94206 -6.00996
------------------------------------------------------------------------------
generate h3 = honors
replace h3 = 0 if lang<50
replace h3 = 1 if lang>=50
logit h3 lang, nolog
lang > 48 predicts data perfectly
logit h3 female lang, nolog
lang > 48 predicts data perfectly
Categorical Data Analysis Course
Phil Ender