Daryl says that they have to forecast tornadoes before they start hitting "places they have never hit before such as Chicago, Los Angeles, even London." The Chicago and London area have been hit numerous times by tornadoes over the years, and the L.A. area has also had some tornadoes albeit usually smaller and weaker ones or waterspouts that move inland. A seasoned storm chaser would have known this and used better examples.
When Gary and Trey are driving into town the tornado is over a block away in front of them. Then when their SUV spins out they appear to be right in the middle of the tornado, but after they crash and head for shelter in the bank the tornado appears to be over a block away again.
When the buses and cars are fleeing the EF-5 tornado, in one shot out the windshield of the bus Gary is driving shows he is following other buses. After one brief interior shot the next shot out of the windshield shows only cars ahead and no buses although you can see far down the road.
The tornado that hits downtown Silverton dissipates about a block or so from the Titus and the bank where the people are hiding out. But somehow there is severe damage further down the street from the Titus and the bank where they tornado did not hit, including a flattened building and Gary's SUV being flipped over.
When the two tornadoes spin up in the downtown scene with the amateur storm chasers, the second tornado immediately contains debris, presumably from scouring the asphalt, a tornado that newly spawned and of that magnitude would not have scoured asphalt. For a tornado to scour asphalt, it would typically be rated EF4 to EF5 (especially EF5) on the Enhanced Fujita scale.
During the final storm scene when the tornado is pulling on the armored vehicle, they connect it to the Semi. The 300 mile per hour winds can pull the semi through the tunnel OR the wire would snap and injure one of them and send the armored vehicle flying.
When the first tornado to hit the Silverton area forms, Allison waits until it is on the ground for several minutes before calling in the tornado warning. Any professional storm chaser would have called the warning in right when the wall cloud formed, or at the very least at the first sign of the funnel as public safety would have been more important to them than observations or photos/videos.
Considering this takes place in Oklahoma, residents would take seriously any threat of massive tornado threats. If the meteorologist in the van knows it, surely the weather bureau would know it too. There is no way the school would plan an outdoor commencement ceremony on an afternoon that offered such potential for extreme damage.
The goof items below may give away important plot points.
When Gary is performing CPR on Donnie, he does it incorrectly. He does 23 chest compressions, and then 3 rescue breaths. Current Red Cross guidelines call for 30 chest compressions and only 2 rescue breaths. The hand placement is also wrong; Gary had his hands on Donnie's upper chest but CPR is performed about 2 finger widths above the lower tip of the sternum(breastbone). While this could be considered stress due to the moment, his late ex-wife was a nurse, and Donnie himself said that they learned CPR from her.
After the bus driven by Gary is cut off from the other escaping buses by the falling power lines, Allison gets out to check on the people in the cars that went off the road. The two cars ahead of the bus appear to be empty. Allison runs to only one car and asks them to get into the bus, because their car is stuck in the mud. But there are already dozens of footprints in the mud by the bus' door before Allison even leaves the bus.
When the final tornado is hitting the high school and the school bus and cars get blocked by the falling high voltage lines, Allison says they have to "go back and find another road." From the overhead views of the road, power lines and tornado, there was no other road between them and the tornado visible.
The people riding out the F-5 tornado under the storm drain could not have survived because the sustained winds of over 300 miles per hour would not have allowed any of them to breathe, and the visible debris shown would have ripped the flesh from their bodies.