Night Nurse, starring Barbara Stanwyck. Some kind person has uploaded the whole thing to YouTube. [Edit: unfortunately, this video has been removed. I encourage anyone interested in good things to seek out and enjoy this film.]
"In a big way, sister."
The curious transatlantic accent explained. I've often wondered why people in black-and-white movies spoke like this, but somehow or other I thought it had to do with being from Connecticut.
Grant and Hepburn demonstrate it nicely in this clip from "Bringing Up Baby":