Why does increasing the number of bootstrapped cases make PLS coefficients significant? I am running a PLS model with a low number of observations ($n=50$). While several pieces of academic work argue that this sample size is appropriate to run this type of model, I am quite confused when it comes to the number of cases I take in my bootstrapping procedure.
I get some very strong path estimates for certain paths without a significant t-statistic (50 cases, 500 samples). However, when I put the number of cases above n (lets say 100 cases, 500 samples), the t-values are more in line with the path estimates.
 A: Bootstrapping typically involves sampling with replacement from your sample data. The size of each bootstrapped sample should be the same as your actual sample size (i.e., n=50). This allows you to study the sampling distribution of your estimators in your actual sample size. 
As @Andy notes, you can arbitrarily increase your bootstrapped sample size (lets call it $b$).  As $b$ approaches $\infty$ the standard error of your estimator will tend towards zero. All non-zero coefficients will by definition be statistically significantly different from zero. I remember reading a journal article where an author artificially inflated their sample size using bootstrapping and reported the resulting p-values. This is very wrong! It also may help to explain some people's naive mistrust in the concept of bootstrapping.
Bootstrapping is not a tool to make standard errors smaller. It is a tool to, among other things, give an estimate of standard error. 
Your sample size is small (i.e., n =50) so presumably you have larger standard errors than you would like. This is the reality. Thus, you can accept the large standard errors or collect more data and increase your n.
