EDISON, N.J. — Neetu Singh was headed for the trash with a diaper from her 4-year-old son. She made her way from the family room to the back porch of her home in this Middlesex County suburb.

The lights were off outside, and beyond the family’s corner lot and its six-foot-high fence, the neighborhood was still.

Then as Ms. Singh started to slide open the glass door, she was set upon, overpowered by three assailants who sprang from the darkness. The men, their faces almost completely sheathed in black knit caps, burst in waving guns. “Be quiet. Where is the money?” Ms. Singh said they demanded.