Multiple Regression - Adjusted Rsquare & SEE vs. coefficient sig.

I'm doing an assignment where we are just learning about regression. We were given a set of data and ran it through SPSS. Now, I am trying to interpret it. Hopefully I make some sort of sense to people who know more about it and that my wording is OK. :)

At this point, basically I am comparing whether adding in more and different independent variables better predict the score of a dependent variable. I started with two IV's and found the R squared adjusted to be .312 with a 5.019 SEE. In the coefficients box, both IV's were statistically significant predictors (looking at the t value and sig value at .05).
In the next exercise, I added a third IV to the two existing IV's. Now the R squared is .453 and SEE is 4.477, which I see as a "yes, this does better predict the DV" since more of the variability can be attributed to the IV's and the error rate decreased. However, the three IV coefficients are now NOT statistically significant predictors. So, does this mean that this is not a better predictor at all? Or am I going more by the R squared adjusted and SEE?

I ask because my professor had us change variables and add some more a couple more times and in all of the cases, there are either none or fewer statistically significant coefficients for the IV's but he keeps asking whether it is a better predictor or not. In each case, the R squared numbers continues to increase while SEE decreases, which I think is what he was wanting us to talk about.
So, if the coefficient is more important than the R squared and SEE, then my answer is going to be that the first regression run in which all of the IV's were statistically significant predictors is the only one that is accurate? In that case, I'm kind of unsure why he had us run so many rounds of it. :) So, I think I'm not understanding something. I think maybe it's not a "vs." situation but go by the R squared and SEE first and then the coefficient significance gives a "boost" but isn't the decider? Not sure here.

Hopefully this makes some sort of sense. I am obviously not well versed in this..first attempt at regression stuff. So..I am sure I'm talking about things strangely but hopefully someone well versed can figure out what I mean anyway :)

Thanks SO much! Just learned about the site from a fantastic is this? Thanks! :)
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Nevermind, got an answer from my professor. Seems that this is a conundrum of regression and generally people end up going by the R Squared and SEE even though it would be best to have good numbers in those AND stat. sig. coefficients.