Hi Dave,

Genesis 6:4 in the New American says it this way:

At that time the Nephilim appeared on earth (as well as later), after the sons of Heaven had intercourse with the daughters of man, who bore them sons.
They were the heroes of old, the men of renown.

(NAB) New American Bible

Sounds like your reading from the King James which states:

There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.

(KJV) King James Bible

The Nephilim were an antediluvian race (pre-flood) race which are referred to in the Bible as giants.

This is what my Catholic Commentary on Holy Scripture states for Genesis 6:4:

The Nephilim were upon the earth in those days, and also after that, when the sons of God ..."

The Nephilim are mention in Numbers 13:34 (Masoretic Text 33) as a Canaanite race of giants in Moses' time; hence the clause and also after that, which is in origin a marginal gloss. The meaning and etymology of the name are unknown. They are referred to in Wisdom 14:6, Ecclesiasticus 16:8, Baruch 3:26-28, etc.

The term giant does not imply more than unusually tall stature, and its import is relative to the average height of the race using it. The Hebrews were not a tall nation. Tradition of old preserved stories of their exploits but these have long since perished.

Note: A gloss is an explanation or translation, by means of a marginal or interlinear note, of a technical or unusual expression in a manuscript text.

Hope this helps,

Mike