As to whether one can have personal morality without religion, the answer seems clear enough to me: yes; I know lots of atheists and agnostics who are generally morally upstanding people. ISTM the questions that follow, however, are, what sort of morality(-ies)? and, are some moralities better than others? and, why does it matter?
First off, it doesn't seem to me a coincidence that most morally upstanding atheists living in Christian/post-Christian societies tend to espouse basically Christian morals. Why is that?
How does the atheist justify adherence to one morality amidst a multiplicity of possible moralities?
The last question, why does it matter?, is really just Ivan Karamazov's statement 'Without God, everything is permissible' in question form. An atheist certainly can be moral, but there doesn't seem to be any absolute reason why he should be; and without a should/ought, morality is dead.
Without God or the gods, each man must necessarily be his own moral arbiter. It is not enough to say, let the community decide: traditional societies have, rightly I think, always connected the community with the gods: 'our gods' defines 'our community'. The atheist society is necessarily a society of individuals, not a community.
As to public morality, which is to say, as to whether we can have a cohesive and just society, without religion, I would say that seems more difficult: on that question the historical verdict seems more likely to be in the negative.