All things being equal, as sample size increases the P-value will decrease, even when the correlation coefficient is the same. For example, if n=20 and r=0.10, the P-value may be > 0.05 (typical significance level), but if n=20,000 and r=0.10, the P-value might well be < 0.05. This isn't limited to correlation coefficients, P-values generally get smaller as sample sizes increase. This is why in some fields overemphasis of P-values is somewhat frowned upon.