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We'll need more information.
What phone do you have?
Which app are you using when you try to save these contacts? Most devices have at least two contact apps Google Contacts and either the manufacturer, like Samsung, or one from the carrier, which one are you using?
How are you trying to save these contacts? Are you dialing the number in the Phone app and then using the (+) add icon to create a new or edit an existing contact or are you trying to add some from text messages or are you doing something else, some other way?
Are you using a third party launcher like Nova?
What happens if you edit a contact, do the changes stay?
How long has this been happening?
How long have you had the device?
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