limit LED current only with transistor and base resistor
Short answer, no. You need to either have a resistor in series with the LED (easy), or design the driving transistor to deliver a constant current (harder).

Long answer, well yes, if you want to go to that trouble. If you use a large base resistor (I'd expect to see more like 10k/100k rather than 250 ohms) then a limited base current, multiplied by the hFE of the transistor, should give you a constant collector current. BUT hFE is not a well specified parameter. If you buy a cooking grade transistor, there might be a 6:1 spread of hFE. If you buy binned versions, so a -A or -B versions, then you might get down to 2:1 variation. Unfortunately, hFE varies with temperature, collector voltage, and base current as well. 

Given that LEDs have quite a wide current operating range, a LED that's happy with 10mA can still look fairly bright at 1mA. If you select a resistor for each transistor to get your target current, then you could get reasonable subjective performance. You're trading selection time and consistency for ease of build.