It is possible to cut paper thin wood strips on a bandsaw, appart from the blade type, the wood species would have also significant importance.
But the question is, how wide and how long, do these strips have to be...???
As a single example, if you have a board 2" thick x 10" wide and 4' long, cutting thin strips on the 2" thickness, is not a big deal but, if you want to cut thin slices from the 10" width, and there there, running the board upright on the bandsaw, those are 2 very different situations...! which one is it...??
PS: A proper bandsaw is known to have the most number of adjustment points than any other wood machine, and not everyone is qualified to to a full calibration that unfortunately may not last long, a simple change of blade can throw many things out...!
Most of those calibration/adjustments points, are something that most bandsaw users don't even know exist however, there a few of us that can pull them appart in 100 pieces, know where every piece goes and most importantly know what it does, this the very reason why we/those people can relate the problem to its origin and make the necessary adjustments however, and this should be said, some of more economical/cheap bandsaws, particularly the smaller size ones, have less adjustment points/possibilities, and its accuracy depend considerably of the quality and calibration of each of the pieces put together by the manufacturer, you can get a fairly good one and you can get one with factory faults that, you will never be able to cut straight with, and that is just a fact...!
Cheers
George