Thirty-nine people survived the crash, although dozens were seriously injured.
No other vehicles were involved in the crash, in which the bus veered into an overpass support column on the A4 motorway near Verona.
The bus was returning to Budapest with students aged 15 to 17 who had been on a mountain holiday.
The impact of the crash shortly before midnight on Friday was so strong that some passengers were thrown out of the vehicle.
It is understood the bus burst into flames shortly afterwards.
Sixteen badly burned bodies were pulled from the wreckage. Of the 39 survivors, 26 were injured, some seriously. The 13 who did not need medical attention were taken to a nearby police station.
Hungarian foreign minister Peter Szijjarto told reporters in Budapest: “One passenger is currently in an induced coma and in life-threatening condition.”
Details of the cause of the crash remained unclear, but Szijjarto said the bus driver lost control of the vehicle, which hit a guard rail and then the overpass support before catching fire. Investigators found no brake marks at the scene.
There were also reports that a Slovenian lorry driver who was travelling behind the bus had noticed a problem with one of its wheels and tried to alert the driver.
According to Italian news agency Ansa, Judit Timaffy, Hungary’s consul-general in Milan, said that a number of the pupils were saved by a sports teacher, who returned to the middle of the fire to drag them out. The teacher suffered burns as a result.
In Budapest, the foreign ministry said official information was that there were 54 passengers, including adults accompanying the students, and two drivers aboard, but it believes the actual number was higher – for reasons yet to be determined.
The bus crash caps a tragic week for Italy in which an avalanche buried a hotel in the Rigopiano resort in the Gran Sasso mountains.
Emergency crews yesterday continued their search for more survivors in the Hotel Rigopiano after pulling out four people from the rubble overnight.
Nine people have so far been found alive in the rubble following Wednesday’s avalanche. Around 30 people were in the hotel at the time.
Two other people escaped the devastation just before the avalanche struck, including Giampiero Parete, a chef holidaying with his family who first sounded the alarm by calling his boss.
He was reunited with his wife and two children yesterday after they were among the first to be located and extracted from the debris.
Firefighter spokesman Alberto Maiolo said four bodies had been found in the rubble, in addition to the four people pulled out alive overnight.
“We are trying to verify if there are other people and when we will be able to pull them out,” he said.