SAN FRANCISCO—When Alexandra Goldman got notice of her rent increase—from $6,000 a month on a five-bedroom house she shared with roommates to $11,000—she says she was in disbelief.

"We knew immediately we were not going to be able to pay that much money to live there," said Ms. Goldman, a 28-year-old planning consultant whose share of the rent was about $1,000 a month. After receiving the notice in October, she said the house's occupants ended up dispersing to other rentals. She is living with other roommates now...