It really depends on the bank. The big national banks usually have more strict regulations about it, while the smaller, independent banks are generally more loose about it. But really, just be nice to the tellers and those that save out coins/bills might save some out for you. Of course, if all you do is order boxes (or dump), they aren't going to be wanting to save out coins for you, but some cookies or chocolates goes a long way. Heck, even just talking to cherry picking tellers about coin collecting you can make a new friend in the hobby.
Really, I don't see this hatred on the forum for people who deal with coins buying silver coins that they come across. If any of us were in that position, I can guarantee you that every one of us would be a "cherry picking teller", and most likely worse than just simply looking for silver rims, some of us would search for die varieties and the whole nine yards. And really, it is kinda silly that we expect tellers, especially those who know about silver, to simply give up any silver that might come through their drawers to people, in many cases, who they have never seen before and will never see again. If you really get to know your tellers, they will be nice to you, but you have to realize two things:
A) Very little silver really comes through in a bank. Most loose change is dumped into a machine, and the goal of running things through a machine is speed and unless you happen to notice a merc or a walking liberty, most of it is put into bags and sent back to Brinks/Loomis/etc. Teller trays only hold about $10-20, or about 3-4 rolls. Unless you are in a really busy bank that deals with a lot of small transactions, it might take a week or more to get low enough in a tray to need to crack open a new roll. A single teller in the course of a normal month might handle only a hundred dollars or so in loose, search-able change. Really, the average CRHer goes through more change in a night of searching than most tellers. And old ladies who cash in their late husband's coin collection is a very rare occurrence indeed.
B) Handling large amounts of change is annoying. While ordering boxes and such isn't too bad, dumping large amounts of coin is a pain because changing bags in many machines is a pain. If you don't fill a bag, dumping with a machine is fairly painless, but on many machines, bags, especially the half bags, are awkwardly located and hard to put the new bag on the nozzle that the coins come out of.
So bottom line, be nice to your tellers and they will be nice to you. The cherry picking teller might just be your new best friend in the coin collecting hobby, after all, you might have a few dates the teller wants (such as a 1970-D) and she might have some generic silver you want (like some 90%ers).
