Torrential rain overnight left many roads impassable, while rail services were cancelled or delayed.
The Environment Agency issued five flood warnings across the county.
Warwickshire Fire and Rescue said it had rescued a number of people trapped in their cars due to rising floodwater.
More on this story and others from Coventry and Warwickshire
Network Rail said flooding on train lines around Daventry, Rugby and Wolverton delayed services between Birmingham New Street and London Euston.
Trains on the West Coast Mainline also ran at reduced speeds due to heavy flooding.
Rose Plumbley, headteacher of Leamington Hastings C of E Academy in Rugby, said many pupils and staff had not managed to get into school.
"We're in the Leam Valley and it's flooded on both roads into school," she said.
"Some pupils did manage to get in but it became very clear that many weren't going to make it in. We decided to close so that the children can get home in daylight safely."
Firefighters rescued drivers from stranded vehicles in Great Alne, Wolston, Baginton and Princethorpe.
There were also reports of cars stuck in Kenilworth Ford.
There were flood warnings at the River Dene at Walton, River Sherbourne at Allesley, River Stour at Mitford Bridge, Shipston on Stour and Tredington, River Stour from Halford to Clifford Chambers and River Alne at Bird in Hand.

Speaking from South Africa, Mr Machar told the BBC that his rebel faction could still negotiate a peace deal with President Salva Kiir.
His statement comes despite last week's heavy fighting in the city of Malakal.
Mr Machar, who first fled to the Democratic Republic of Congo, is now being treated in Johannesburg.
In July, Mr Machar's bodyguards and President Salva Kiir's presidential guards fought each other, sparking days of violence.
Hundreds of people died and more than 100,000 fled across the border.
The fighting erupted less than a year after a peace deal was signed to form a unity government and end the civil war.
Riek Machar profiled
Speaking to the BBC's HARDtalk programme, Mr Machar said: "I'm going to return to South Sudan."
"Because President Salva Kiir doesn't want democratic and transparent and fair elections to be conducted, he attacked us, he has restarted the war.
"But I am hoping that wise leaders in the region, and in Africa, and the rest of the world will throw up a political process which will bring about peace again, and the resuscitation of the peace agreement, and the reconstitution of the transitional government of national unity."
Mr Machar also denied that claims that he was a warmonger, saying that his troops were simply defending themselves against government attacks.
President Kiir's officials accuse Mr Machar's forces of starting the attacks.
Both sides in the bitter conflict have been accused of perpetrating atrocities, the BBC's Karen Allen in South Africa reports.
More on South Sudan's crisis:

The Iridis4 has state of the art coprocessors which can each perform a trillion calculations per second - a measurement known as a "teraflop".
The university said the new machine would allow academics to work on more projects at faster speeds.
Iridis4 will be used for subjects such as engineering, archaeology and medicine, as well as computer science.
Pro vice-chancellor Prof Philip Nelson said: "Staying ahead of the game in high performance computing [HPC] is vital to help the university stay competitive.
"Simulation and computation enabled by HPC are recognised globally as the third pillar of modern research and this investment will ensure we remain world leaders in this field."
Iridis4 will be used for a range of research, including engineering, archaeology and medicine, as well as computer science.
The world's most powerful computer is China's Tianhe-2, which can perform 33,860 trillion calculations per second.
The university said its new computer ranked among the top 10 in the UK.
The most powerful is at the Science and Technology Facilities Council in Warrington.
Others are based at the University of Edinburgh, the European Centre for Medium-Range Weather Forecasts and the United Kingdom Meteorological Office.
Fourteen schools were forced to close due to severe flooding in Warwickshire.
[NEXT_CONCEPT]
South Sudan's sacked Vice-President Riek Machar - who fled the country in August - has vowed to return, saying his credibility is intact.
[NEXT_CONCEPT]
A £3.2m supercomputer, one of the most powerful in the UK, has been installed at the University of Southampton.