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Did they tell you that they overfilled it? Does the oil level on the dipstick show above the line on the dipstick? If it is a quart over and you only drove it few miles, there should be no real harm done, but I wouldn't leave it a quart over.
I would first go back to the dealer and have them re-do the oill change. They should have known what the correct amount was to put in. That's assuming the "technician" can tell the difference between the numbers 5 and 6.
If you can't get back to the dealer, you may want to remove the oil filter and replace it with a new one. The filter should contain about 1/2 quart and bring the level down to where it should be OK. Have the engine warm and let it idle until the pressure gauge shows a normal reading.
There's not much damage an extra quart will do.
I wouldn't worry about it at all.
Transmission a quart over I'd worry about.... engine "no worries".
Both my Vettes will use about 1/4 of quart between 3000 mile oil changes.
I typically overfill them that 1/4 quart so they are normally sitting at the full level when it's time for the oil change.
Don't worry about it, there's nothing it's going to hurt.
If they installed a new oil filter, it takes about 3/4 qt and you are over by only 1/4 qt. Even if they didn't install a new filter, 1 qt overfill won't hurt. Overfilling is damaging only when the oil level is so high that the crank hits oil which stresses the crank and worse, foams the oil. Foamed oil delivered to bearings means no more bearings. One quart won't do it. I would siphon a qt out of the dipstick tube. When I change my oil, I fill the oil filter too , then install it, but I have never heard of any commercial places that do this. I don't want my engine turning without oil being pumped.
Chevrolet used to recommend that you put an extra quart of oil in the LT4's.
Before anyone flames me I need to state that it was for auto-x or high speed turns etc. For normal street driving it was not necessary.
Seems that the higher revving LT4 combined with the cornering abilities of the C4 chassis could contribute to an oil starvation problem.
So, the good news is that wont hurt the engine. :)
The bad news is that you need to find a different service technician. :mad