The phrase "1,000 years is but a day" allows people to connect the existence of the world to the seven days of creation. Thus, each millenium in the existence of the world is the equivalent of a "day" of creation, with the seventh millennium (6001 - 7000) being the equivalent Shabbat. Since the year 6000 is the last year of "Erve Shabbat", the mashiach would have to come then for the "spiritual" millennium to begin. An example of this is found at What is the significance of the year 6000 in the Jewish calendar? Babylonian Talmud Rosh Hashana 31a and Sanhedrin 97a
The Talmud tells us that this world, as we know it, will last for six
thousand years, with the seventh millennium ushering in the cosmic
Shabbat, the Messianic Era. Six days a week we work, and on the
Shabbat we rest and enjoy the fruits of our labor; the same is true
There are people (who are recognized as "normative") who seem to take this literally.
For example the Vilna Gaon is supposed to have taken this literally and made this claim (I do not have a trustworthy link for this).
Moshiach and the World Today by Rabbi Peter Winston appears to take this literally.
The first thing to know is that history will last only 6,000 years
(Talmud - Sanhedrin 97a). This is because the six millennia are based
upon the six days of creation, as hinted to in the following verse:
For one thousand years in Your [God's] eyes are but a day that has
passed. (Psalms 90:4)
Rebbetzin Jungreis: By the Year 6,000, Mashiach Has to be Here has also appeared to take this literally.
"Listen carefully, friends, to what I'm telling you. Hashem, Elokei
Yisrael, created this world that we are living in today in six days.
Every day was a thousand years. This world, as we know it today,
cannot last beyond 6,000 years. Right now, we are in the year 5769,
which means it's Erev Shabbos of the world. By the year 6,000,
Mashiach has to be here. He could come much earlier. But by the year
6,000, he has to be here. … the Vilna Gaon said that the last war,
Milchemet Gog uMagog, is going to last only 12 minutes because they
are going to have such weapons…. We know that the final redemption,
the final Geula, it's going to be like when you left Egypt – only
one-fifth of our people left Egypt. Four-fifths perished… during the
plague of darkness.
"So I'm appealing to every Jew. Every negative prophecy can be
changed. We can bring Mashiach today. Right now, we are living in a
period called Erev Shabbat. It's Erev Shabbat, because when Mashiach
will come, it will be the day that will be all Shabbat, the seventh
day….Let's bring Shabbos early, and let us to bring Shabbos with
menucha [ease], with shalom [peace], with simchah [happiness] – Is it
possible? Absolutely?! Every negative prophecy can be changed."
While it can be treated literally, we do not know for sure exactly what will happen, but we should act as if our actions can cause the Mashiach to "come earlier" (like lighting candles earlier Friday afternoon) or to delay it. Consider that by miscounting one day at Sinai, the Bnai Yisrael caused the sin of the Golden Calf. We should not be like those who expected the mashiach on a particular day and then dropped everything as an over-reaction to having been wrong.
We should also be careful as to what we expect to happen when the mashiach actually comes (bimheira beyameinu).