For awhile, I used to think:  
 If $depth\ M\ge depth\ N$ then $depth\ M_p\ge depth\ N_p$; for any prime ideal $p$ and finite R-modules $M$ and $N$ ([Which is not true](https://mathoverflow.net/questions/224135/assuming-depth-m-ge-depth-n-what-can-one-say-about-depth-m-p-and-depth-n-p)).