My take on the NTG3 - it's a fantastic mic. I've never used the MKH 416 but I also checked all the comparison material and opinions online and yeah, it seemed close enough.
The only thing with the NTG3 - you have to be aware that it's quite bassy, and it doesn't have a low cut filter. I personally don't have any problem with that, but I know some people who didn't like the mic because of that. I don't know if the MKH416 has a low cut filter, but I believe it does?
Anyway, the way I see it is that a lo cut filter is mainly useful for dialogue recordists. I only really record sound effects so I don't mind at all - can fix things in post if really necessary, and I don't have to run around with a boom pole, worrying about low frequency rumble from handling or cable noise.
However, if you have a bit of extra budget... May be worth scanning eBay for a MKH60. It's really a big step up from the NTG3/MKH416 in terms of smoothness, natural representation, transparency. It's more expensive - but on eBay I see it go sometimes for a little more than a MKH416, and in this case, the price difference is most definitely worth it over the NTG3!