Does one need to have all the genes for bipolar disorder in order to have the disease?
No. In genome-wide association studies (GWAS) of psychiatric conditions, it is a constant finding that the associated variants each influence the probability/risk. The effects of all known variants can then be summed up to a polygenic score (PGS). A person with higher PGS is more likely than someone with a low PGS. But some people with low PGS (i.e. people who don't have "all the genes") still get the condition, it is just not as frequent as people with high PGS.
Another reason why we can say 'no' to this question is through findings of quantitative genetics studies. These estimate the heritability of certain traits. For the answer to be 'yes', the heritability would have to be one ($h^2 = 1$). However, as far as I know, there are no known psychiatric conditions that have heritabilities of one. Some high (e.g. $h^2\approx 0.8$), yes, but not one.
It's also worth mentioning that there is probably no sharp line between having bipolar disorder and not having it. It's a medical diagnosis that is useful but rough. Most likely, all of us have a little "bipolar" in us. I think it's probably most useful to view it as a spectrum.
Is it possible for a person to have all the genes for bipolar disorder and still not have the disease?
This is not known and it may not even be a well-defined question. We don't know all the variants that affect the risk of bipolar disorder (or most other polygenic disorders). So we cannot say for sure.
Some 'omnigenetic' hypotheses propose that all genetic variants have some effect on all traits. Some effects are just smaller and some larger. In this case, the expression 'all the genes for X' may be technically meaningless.