What's actually happening when the door stops short
Your opener is a small motor. On most homes it's rated 1/2 to 3/4 horsepower, and it was never designed to lift the door by itself.
The springs do the heavy lifting. A torsion spring above the door counterbalances almost the entire weight so the opener only has to nudge it along.
When that spring breaks, the balance is gone. The opener strains against 200-plus pounds, moves the door a few inches, hits its safety limit, and stops (or reverses).
That's why the symptom is so consistent. A few inches, a groan, then nothing. It's the single most common garage door failure we see.
The most likely culprits, in order
Start with the springs because that's where the money is. A broken torsion spring accounts for the vast majority of these calls.
Walk in and look at the shaft mounted above the door opening. A broken torsion spring shows a clean 1 to 2 inch gap where the tight coils separated.
If you have extension springs instead (long springs running along the horizontal tracks), look for one hanging loose or a cable dangling.
A spring doesn't have to snap to cause this. A fatigued spring that's lost tension over years leaves the door heavy and the opener struggling past the first foot.
Less often it's the opener itself, with the up-force set too low or a worn drive gear. And occasionally it's mechanical: a bent track, a roller out of its channel, or ice welding the bottom seal to the slab.
The hand-lift test sorts most of this out in thirty seconds, which is why it's step one of any real diagnosis.