Police and school officials swept Lake Worth High School after a bomb threat.
Parents in Arlington learned of a threatening note targeting two west-side schools.
In Haltom City, a Birdville Elementary student reported hearing possible gunshots in the parking lot, bringing officers from three cities and prompting a one-hour lockdown.
Police finally determined that the sound was likely a vehicle backfiring.
Nerves are raw after Friday's massacre at Sandy Hook Elementary School, especially among parents, teachers and school administrators. The tragedy in Connecticut has rocked the nation's sense of security in a way not seen since 9-11 or the attack on students at Columbine High School in Colorado.
"I think many of us are very much on the edge," said Alex del Carmen, chairman of the University of Texas at Arlington's department of criminology and criminal justice. "I think this is going to live in our long-term memory.
"We fear a copycat."
School and law enforcement officials say they always take threats seriously, but they acknowledge that the shootings in Newtown, Conn., have set parents on edge and prompted reassessments of emergency plans.
Michelle Boykins, spokeswoman for the National Crime Prevention Council, said people typically emerge from devastating tragedies with a deep sense of alertness. The nonprofit Virginia-based organization has promoted crime prevention for 30 years.
"We think it is not necessarily a bad thing that people have a heightened sense of awareness right now," Boykins said.
"As a resident in the community, you need to be vigilant."
'Pray for Birdville!'
About 8 a.m. Tuesday, someone called 911 to report possible gunfire near Birdville Elementary School. The school and the nearby Shannon Learning Center were locked down as police from Haltom City, Watauga and North Richland Hills responded.
"Nobody is hurt," Haltom City Cpl. Joe Hackfeld reported after officers surveyed the campus and determined that a parent's vehicle had backfired.
Police said they responded to the potential threat in the same way they would have if the Connecticut shootings hadn't occurred.
"We always take the safety of the children with the utmost importance," said Keith Bauman, a North Richland Hills police investigator.
Seeing the heavily armed officers at school upset parents and relatives of students, some of whom said they are still struggling with their emotions after the Sandy Hook tragedy.
"After what happened in Connecticut, it's scary," said Martha Garriazo, a grandmother who took her 6-year-old granddaughter out of school after police cleared the campus.
The reaction on social media was swift and perhaps premature. One post appeared to sum up the prevailing sentiment: "Pray for Birdville! Spread the word."
In Arlington, school officials increased security at Young Junior High and Miller Elementary after the discovery of a note containing an alleged threat to teachers and staff members.
On Monday, the district sent a letter to parents explaining the increase in security personnel and uniformed police officers at the campuses.
"There was a note found on the property of Young Junior High School," district spokeswoman Amy Casas said Tuesday. "We were notified by police and immediately started working with them to try to see if they could determine who could possibly have written it."
Police determined that there was "no imminent threat," the district said.
At Lake Worth High School, a staff member found a note Tuesday suggesting that a bomb might be in the school, said Christy Stinson, the district's executive director of business and support services.
Police and fire authorities were called, and students and teachers were evacuated to a nearby store parking lot.
The school was declared safe at about 8:30 a.m.
Stinson said educators are trying to ensure that students and parents feel secure.
"They are frightened, very fearful for their children," she said.
"I'm a parent also. ... I understand their fear and their concern."
Lake Worth will have more police on campus Friday, the one-week anniversary of the Sandy Hook shootings.
"Everyone is just very, very concerned," Stinson said.
'An extra sensitivity'
In Stephenville on Saturday, graduation ceremonies at Tarleton State University were delayed after a threat was reported.
In Granbury, district officials are dealing not with threats but with rumors of threats.
"Unfortunately, several rumors and speculative thoughts have been posted and reposted on social media. This hinders the investigation and does not help us or the police get to the bottom of any possible allegation," Superintendent James Largent said in a statement posted Monday on the district's website.
Emergency planning is ongoing, leaders of area districts say.
"We do whatever we need to do based on the threats that exist," said Clint Bond, spokesman for the Fort Worth school district.
Bond said each Fort Worth campus has an incident commander -- the principal or someone else.
In an emergency, the incident commander assesses and responds to the situation.
Bond said he wants parents to be active members of emergency planning.
Tips and resources are available on the district's website, and students are encouraged to discuss the drills with their families.
"If somebody sees something, they should say something," Bond said. "Our biggest tool is the vigilance of everybody involved."
Arlington school administrators have been in discussions with police and district security since Friday to determine how to enhance security, said Casas, the spokeswoman.
The district requires that campuses regularly conduct drills, she said, so teachers and students alike are prepared for emergencies.
"We understand, given what happened, that everybody's going to have an extra sensitivity to this," Casas said.
"This is something that's going to be an ongoing discussion."
Staff writers Shirley Jinkins, Patrick M. Walker and Susan Schrock contributed to this report.
Diane Smith, 817-390-7675