I think it's important to understand why there is a difference between the NLT and the other translations.
The NLT began life in 1962 as Ken Taylor's paraphrase of the NT letters. Almost a decade later, in 1971, Ken Taylor's efforts had spawned The Living Bible, a PARAPHRASE of the entire Bible, based on the ASV. His intent was to make a version of scripture that was readable and understandable by children.
In 1989 it was decided, by whom I couldn't discover (perhaps the publishers of TLB), to produce a TRANSLATION based on Taylor's approach. In the Preface to the NLT, you will see that it was a work of ninety scholars over a period of seven years, to produce a thought-for-thought translation of Scripture for "the average reader of modern English ... at the reading level of a junior high school student."
The purpose and process of translating texts in this way is unlikely to produce ambiguity. The NLT gives readers the consensus opinion of the scholars involved in making a decision about what is being communicated by the text, which, of course, will cause issues for those whose theology depends upon another branch at any point of possible ambiguity.
My conclusion is, therefore: YES, in the consensus opinion of the scholars who produced, reviewed, and refined the NLT translation, the original Hebrew does support this interpretation.
Having said that...
You will find a typical interlinear presentation of Genesis 6:3 at Bible Hub, which results in the ESV's:
Then the Lord said, “My Spirit shall not abide in (fn: contend with) man forever, for he is flesh: his days shall be 120 years.”
and the NIV's:
Then the Lord said, “My Spirit will not contend with (fn: remain in) humans forever, for they are mortal (fn: corrupt); their days will be a hundred and twenty years.”
And here's my alternate rendering:
וַיֹּ֣אמֶר יְהוָ֗ה לֹֽא־ יָדֹ֨ון רוּחִ֤י אָדָם֙ עֹלָ֔ם
.always in-man My-breath will-remain not the-LORD Then-said
בְּשַׁגַּ֖ם ה֣וּא בָשָׂ֑ר וְהָי֣וּ יָמָ֔יו מֵאָ֥ה וְעֶשְׂרִ֖ים שָׁנָֽה׃
years twenty-and one-hundred his-days and-shall-become flesh he-is, Indeed
Then the LORD said, "My Breath will not always remain in man. Indeed, his days (shall become = וְהָי֣וּ ) one hundred and twenty years."
Or as the NLT renders it:
Then the Lord said, "My Spirit will not put up with (fn: will not remain in) humans for such a long time, for they are only mortal flesh. In the future, their normal lifespan (will be no more than = וְהָי֣וּ ) one hundred and twenty years."
My justification for giving וְהָי֣וּ as "and-shall-become" is,
And God said, “Let there be lights in the vault of the sky to separate the day from the night, (and let them serve as = וְהָי֣וּ ) signs to mark sacred times, and days and years,
And God said, “Let there be lights in the expanse of the heavens to separate the day from the night. (And let them be for = וְהָי֣וּ ) signs and for seasons (fn:appointed times), and for days and years,
In both these translations the "let them serve as ..." or "let them be for ..." could easily have been given as "they shall become signs for sacred times/appointed times/seasons and for days and years"
and Genesis 2:24
That is why a man leaves his father and mother and is united to his wife, ( and they become = וְהָי֣וּ ) one flesh.
Therefore a man shall leave his father and his mother and hold fast to his wife, (and they shall become = וְהָי֣וּ ) one flesh.
Conclusion: There is clear justification, based on the Hebrew language resources available, and the testimony of history itself, for the NLT's use of "no more than".
Whether it is a good or bad English rendering is totally dependent upon whether or not it is in concord or discord with one's personal theology. However, it is abundantly clear that the Hebrew doesn't disallow it.
Based on some feedback in the comments, further justification for giving וְהָי֣וּ as "and-shall-become" comes from the NAS Exhaustive Concordance of the Bible where וְהָי֣וּ occurs as "become" 221 times.
Interestingly, both the ESV and NIV render it in this way when God speaks, in the aftermath of the Deluge, of never again bringing a flood on the world.
I will remember my covenant that is between me and you and every living creature of all flesh. And the waters shall never again become ( וְהָי֣וּ ) a flood to destroy all flesh.
I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become ( וְהָי֣וּ ) a flood to destroy all life.
Since there is abundant support for the use of "become", then we have to consider the implications.
If the lifespan of the people before the flood was typically in excess of 120 years, and God says, "his days shall become one hundred and twenty years", then 120 becomes the UPPER BOUND for the lifespan of man, into the future - that is, "no more than 120 years"
This means I have to amend my previous conclusion, and say now, rather than the Hebrew "not disallowing" the NLT's translation, the stronger claim can be made that there is definite support in the Hebrew for the NLT's translation.