The small overlap test replicates what happens when the front corner of a vehicle collides with another vehicle or an object at 40 miles an hour.
Seven of the 12 tested received "Acceptable" or "Marginal" ratings, while four others received a rating of "Poor", including the Fiat 500L, Nissan's Juke and their all-electric Leaf, as well as the Mazda 5.
The institute concluded they didn't provide enough protection to the occupants' compartment.
"The survival space wasn't well maintained and that means things were pushing against the occupant, pushing on the legs," says the Institute's Joe Nolan.
The manufacturers of each vehicle point out that all four did very well in other crash tests, and plan improvements to do better in this one.
Copyright 2015 Nexstar Broadcasting, Inc. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.