SAN DIEGO—A Rancho Bernardo man who prosecutors say had an “arsenal of weapons” that was accessible to his children—one of whom allegedly threatened to commit a school shooting—has pleaded not guilty to more than a dozen weapons-related charges, and remains in custody Feb. 2.

Neal James Anders, 45, was arrested after Mr. Anders’s 14-year-old son allegedly made threats to carry out a shooting at Rancho Bernardo High School.

Deputy District Attorney Roza Egiazarian said the boy told several classmates “to not go to school on Tuesday, the 30th, because he was bringing guns to school.”

Though a defense attorney for Mr. Anders said Thursday that all of the weapons he owned were in a locked case, the prosecutor said the teen allegedly claimed a lock securing the firearms was broken, allowing him access to the weapons.

Mr. Anders’ son was also arrested, though prosecutors declined to comment Thursday on what charges he may face. A San Diego County District Attorney’s spokesperson said the teen has a hearing in juvenile court scheduled for Feb. 9.

After a gun violence restraining order was served on Mr. Anders’ son, police searched the family’s home and a storage facility, the prosecutor said.

Egiazarian said police found 19 AR-style rifles—including nine unserialized rifles—multiple pistols, magazines, ammunition of various calibers, flamethrowers, armored vests, gas masks, and items that can be used to assemble firearms.

Investigators also found a “Dragon’s Breath” round, which the prosecutor said are incendiary rounds that can shoot “a 200-yard wall of fire.” That round led to a possession of a destructive device count filed against Mr. Anders. He also faces charges for his alleged possession of a short-barreled rifle and unserialized assault rifles.

Investigators also found a rocket launcher and multiple grenades, but those were all inert, according to the prosecutor.

Mr. Anders faces just over three years in state prison if convicted of all counts.

He remains in custody on $300,000 bail.

His defense attorney, Gregory Garrison, unsuccessfully sought a reduction of his client’s bail to $70,000, which Mr. Garrison said more appropriately reflected the counts Mr. Anders is charged with.

Mr. Garrison said his client has no criminal history and was not a flight risk or a danger to the community.

Mr. Garrison said Mr. Anders has strong ties to the community, is employed with a military contractor, holds a “secret clearance” and his employer “anxiously awaits his return to work.”

The attorney also noted that police had seized all of Mr. Anders’ firearms and that Mr. Anders would agree not to possess any other weapons if released.