Dimmers don't turn on at low setting
Many modern dimmable LED bulbs will dim with older dimmer switches, there are some issues. One is that the bulbs may not dim as low as they can, the other is that the bulbs may not come on at the lowest setting. You may need to bring them up and then back down. I've typically had issues with the first while you seem to be having problems with the latter.

New CFL/LED compatible dimmers solve a lot of problems. In addition to being engineered to be more compatible with dimmable LEDs and CFLs they also have a low trim adjustment. Either a sliding switch, knob or some other mechanism to adjust the lowest dimming setting where the bulbs perform satisfactorily.

There are still compatibility issues so it's best to see what dimmers are compatible with the bulbs you have. The manufacturers of dimmers (and usually the bulb manufacturers too) will have a compatibility list so you can get the right dimmer for your bulbs or vice versa.

I got tired of not being able to dim my LED bulbs as low as I wanted so I replaced my dimmers and have been very happy. Most of my bulbs have been Philips and my dimmers Lutron C-L.