Not sure what the problem is but the last email in the inbox was dated 11th June so there are lots missing.I sometimes get a small message failed to sign in.I never sign in to my email account on the phone as all I do is click on the email tab and go straight in.I cannot find anywhere to go to to sign in.The phone is a Samsung Galaxy J3 2016 and I am with Tesco Mobile.
AS JayCee says your account settings have to be correct, the model of phone makes not difference. You should also have a sync tab at the bottom of your email box that allows you to update your emails.
Are you using a hotmail/outlook account or other for your emails?
I found out the problem I had changed my email password on the phone and as I never sign into the phone it was not allowing to to access any of the emails I had received since I changed my password so I changed my password again on my laptop and then changed it on the phone and all seems well now.