Did Grandma give the "wrong" date or place of birth for herself? Did she possibly do it because she actually thought that is where or when she was born?
Keep in mind that on many records where our ancestors provided information on themselves that they were not actually asked for proof. The clerk just wrote down what they gave.
My own Grandma, who would have been 99 today, always gave the same place as her place of birth. Problem is, her birth certificate and other contemporary records give a different location. Grandma just had a misconception about where she was born.
Sometimes errors are actually mistakes, not intentional lies.