Can you apply for a new Russian passport at any time?

This is only a recommendation from the consulates, as the time deadlines for a new passport creation are up to 3 months, so if you want to get a new passport when your old one is about to expire in 2 months, you could get into to situation when you have no valid passport.

The only restriction is that you can't get a new passport without invalidation the old one (Federal Law # 114 in Russian), with only one exclusion: if you regulary go out of the country (you have stamps in your passport for 3 consecutive months), you can create another one, but the expiry date for such passport will be equal to expiry date of the first one.

Update:
About the three consecutive months rule:
Quote from law:

? ??????, ???? ???????????? ?????????? ?????????? ????????? ??????? ? ??????????? (?? ???? ??? ???? ??? ? ??????? ??????) ???????? ?? ??????? ?????????? ?????????? ?????????, ? ?????? ????????? ?? ????? ????? ?? ????????? ???????????????? ??? ?????????? ????????, ??? ?? ??????????? ???????????, ???????????? ??? ?? ??????? ?????????? ?????????? ?????????, ????? ???? ???????? ? ????? ?????? ???????, ??? ???? ?? ?????? ???????? ???? ????????? ????? ???????? ???????? ??????????? ???? ????????? ????? ???????? ????? ????????? ????????.

So you need three points to get the second one passport without invalidating the first one:

• You can't get a diplomatic or working passport
• You regularly passed through the customs. This can be only one way approved: no less than 3 last consecutive months you've already gone through customs (say, one time is a fortuity, two times are coincidence, three times - are you kidding me?).
• You have a job which is related with trips, and your organization can create such request for you.

There are a lot of sites with detailed explanation, like this:
http://www.yfms-passport.ru/two_international_passport

Note: if you are saying that you lost your passport, it will be invalidated, and even if you find it later, it can't be validated back.

In addition to VMAtm's answer, for practical purposes, also consider the field on the passport application where you have to select whether it's your first passport, or whether you're getting it in exchange of a used/damaged/lost one.

9 ????????? ????????

• ?????????
• ?????? ???????????????
• ?????? ????????????
• ?????? ???????????

When picking up the new passport, if the old passport is otherwise valid, it'll be taken momentarily to have holes punched through it, and returned back to you.