Why would an employer not withhold federal taxes?

My wife is a teacher and switched school districts last year.  She filled out her W4 exactly the same way for each; selecting Married filing jointly in Step 1 and noting $4,000 in Step 3 (because of our 2 young children).  However, whereas her old employer would previously withhold federal taxes from each paycheck, her new employer is not.  And her salary is essentially the same.  To correct this, she recently filled out a new W4 and entered a dollar amount to be withheld in Step 4c.   This should mostly fix the issue (and ultimately reduce the amount we'll owe in April), but I am wondering why she would even need to do this in the first place.  Why would her new employer not withhold taxes based on the settings mentioned above - when the old one was doing that.  I think the issue is on the employer's side, but they seem unwilling to admit that.