How long does it take to get internet cable buried?

Depending on the area, 2 weeks to 4 months...

North Central states are not buried till springtime after ground is thawed.

Depending on region may be EMPLOYEE (SE) or contractor (other 12 states).

You can call the ATT Buried Wire Center Monday to Friday during business hours Eastern Time to verify a work order has been created for burial service.

Depending on region may be EMPLOYEE (SE)

If "SE" was to mean Southeast, it's not certain to be an employee in the southeast.  Mine was buried by a contractor using a spade.  My neighbors was buried by machine, and I'm reasonably sure that was a contractor as well.

@JefferMC 

Yes SE is BellSouth 9 state region

The union contract for this district (3) does have a burial tech position ... may not be in all markets. 

The other districts (4 MW-Ameritech, 6 SW-SourhernBell, 9 West- PacTel) do not have this position within work groups. 

For sake of completeness try to cover the possibilities.

@JefferMC   In the louisville Ky market, ATT employees bury our cables. Machine and by hand. Depending on the ground conditions. 

for further clarification of my original question, I am in the Louisville, KY market.  The ground is most definitely not frozen and the cable has been laying there for a month. I have had to install a temporary fence to keep my dogs out of that part of my yard or the puppy will chew it.  

Mine was an AT&T branded utility truck. The guy wore the AT&T reflector vest. He used the spade. Across the street when they buried their line a couple of months ago it was the machine. You do need to call the line bury dept to see if they can update you. 800.924. 9420 Monday through Friday between 8:00 a.m. and 6 p.m. ET.