Yes. It's highly possible you can turn on your old MB with

other power suppply and a charged battery. If your battery not charged and MB detect your other power supply is not strong enough then you can't turn it on.

You may be able to charge your old battery with other power supply. There is a green charge led on dell power light panel area. After it charged (press the button on battery to see at least 3 of 5 led on), then try to turn on MB.