Um, not really.
Underweight is a BMI below 18.5.
Normal is a BMI between 18.5 and 24.9
Obesity is defined as a BMI above 30.
A person who is 5ft tall: normal = 91-115lbs. overweight = 119-138 Obese = 153+ (using the NIH BMI calculator)
So even for a person who is 5ft tall- on the short side - the difference between the upper end of normal and obese is a difference of 38 pounds.
The average height of a newborn is 18-22 inches. For an infant that is 24 inches tall- the upper range of normal is 20 lbs, with obese at 25 lbs.
So, technically you are correct, but 5 pounds difference in a NEWBORN is a big weight difference!
Charts do have some variability, as muscle weighs more than fat. A very muscular person can rank overweight, when really they aren't. so the BMI scale isn't a perfect guide, but saying the range is 5 pounds is rather misleading.
Then again, I think most folks already know all this.