Tuesday is with Mr. Montalvan at all hours. Taught to recognize changes in a person's breathing, perspiration or scent that can indicate an imminent panic attack, Tuesday can keep Mr. Montalvan buffered from crowds or deliver a calming nuzzle. Other dogs, typically golden retrievers, Labradors or Labrador retriever blends, are trained to wake masters from debilitating nightmares and to help patients differentiate between hallucinations and reality by barking if a real person is nearby.

"Tuesday is just extraordinarily empathetic," said Mr. Montalvan, 36 years old, a retired Army captain who received a Purple Heart for wounds he suffered in Iraq. "In bad moments, he'll lay his head on my leg, and it'll be like he's saying, 'You're OK. You're not alone.'"

…At Mr. Montalvan's apartment -- decorated with his Purple Heart and Bronze Star certificates, and pictures and paintings of Tuesday -- they sleep in the same wooden sleigh bed.