High p-values for logistic regression variable that perfectly separates?

I'm using R to run some logistic regression. My variables were continuous, but I used cut to bucket the data. Some particular buckets for these variables always result in dependent variable being equal to 1. As expcted, the coefficient estimate for this bucket is very high, but the p-value is also high. There are about ~90 observations in either these buckets, and around 800 total observations, so I don't think it's a problem of sample size. Also, this variable should not be related to other variables, which would naturally reduce their p-values.

Are there any other plausible explanations for the high p-value?

Example:

myData$FICO <- cut(myData$FICO, c(0, 660, 680, 700, 720, 740, 780, Inf), right = FALSE)
myData$CLTV <- cut(myData$CLTV, c(0, 70, 80, 90, 95, 100, 125, Inf), right = FALSE)
fit <- glm(Denied ~ CLTV + FICO, data = myData, family=binomial())

Results are something like this:

Deviance Residuals:
Min        1Q    Median        3Q       Max
-1.53831  -0.77944  -0.62487   0.00027   2.09771

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)     -1.33630    0.23250  -5.747 9.06e-09 ***
CLTV(70,80]     -0.54961    0.34864  -1.576 0.114930
CLTV(80,90]     -0.51413    0.31230  -1.646 0.099715 .
CLTV(90,95]     -0.74648    0.37221  -2.006 0.044904 *
CLTV(95,100]     0.38370    0.37709   1.018 0.308906
CLTV(100,125]   -0.01554    0.25187  -0.062 0.950792
CLTV(125,Inf]   18.49557  443.55550   0.042 0.966739
FICO[0,660)     19.64884 3956.18034   0.005 0.996037
FICO[660,680)    1.77008    0.47653   3.715 0.000204 ***
FICO[680,700)    0.98575    0.30859   3.194 0.001402 **
FICO[700,720)    1.31767    0.27166   4.850 1.23e-06 ***
FICO[720,740)    0.62720    0.29819   2.103 0.035434 *
FICO[740,780)    0.31605    0.23369   1.352 0.176236
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 1037.43  on 810  degrees of freedom
Residual deviance:  803.88  on 798  degrees of freedom
AIC: 829.88

Number of Fisher Scoring iterations: 16

FICO in the range [0, 660) and CLTV in the range (125, Inf] indeed always results in Denial = 1, so their coefficients are very large, but why are they also "insignificant"?

This is the well-known Hauck-Donner effect whereby standard errors of maximum likelihood estimates blow up. The basic idea is that as the separation becomes complete, the estimate of the standard error blows up faster than the estimate of the log odds ratio, rendering Wald $\chi^2$ statistics useless (and $P$-values large). Use likelihood ratio tests instead. These are unaffected by complete separation.
• brglm implements Firth's penalization, which offsets an $O(n-1)$ term in the bias of MLEs: Firth (1993), "Bias reduction of maximum likelihood estimates", Biometrika, 80, pp 27–38. I'm curious as to why you say "infinite parameter estimates do not present problems in prediction" because that's where, intuitively, they can be most problematic: your model says the probability's 100% for a given value of the predictor on which separation occurred whatever the values of any other predictors. Jun 10 '14 at 11:18