What would cause a soldier to not get promoted to E3 or E4?

The most common reasons are:
APFT flag
HT WT flag
adverse action flag
ADCO flag
commander investigation flag
law enforcement flag
the unit failing to submit the flag lift when it's over
JAG taking 12+ months to review flag lifts
and unit failing to complete the advancement 4187

If your friend does not have a flag, then it's the last reason.

I'm going to go with SGM (Join to see) final reason of the unit not filling out the 4187. This happens a whole lot more than it should on active duty in combat arms units. Most likely, the Soldier was flagged for some reason when it was time for automatic advancement so he didn't get promoted at that time. The good news is that it's a very easy fix. The commander only needs to fill out a 4187 with the correct date of advancement for each rank. The S1 makes the change in emilpo and sends copies to DFAS and the Soldier receives all the back pay

Promotion to SPC is automatic unless there is an underlying issue. I have learned that you really have to take what a person says at face value. Soldiers seldom admit to their shortcomings. But something had to happen to prevent that from happening. I have a feeling that he is flagged for something and he is not telling you.