I'm going to have to disagree in part with the other answers.

When it comes to general experience, such as the 250 hours of flight time or the 100 hours of PIC in certain types of aircraft, those are all cumulative.

But for anything that is a "training" requirement, it has to be at least post-private training. So, for example, if you happened to do a 2-hour, 100 nm night dual cross country to satisfy your private night cross country training requirement, it would not count toward the commercial.

The FAA Chief Counsel even went so far as to say recently that even an instrument rating doesn't satisfy the instrument training requirement in 61.129. There's a huge controversy about that one with even AOPA stepping in to get the FAA to change its mind.

Here's the opinion...

http://www.faa.gov/about/office_org/.../Theriault.pdf

a follow-up reaffirming that private experience doesn't count:

http://www.faa.gov/about/office_org/...011/Murphy.pdf

Here's the AOPA dealing with the instrument training issue:
AOPA Online: FAA clarifies commercial-pilot instrument requirements

I don't think that page requires membership but in case it does:

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FAA clarifies commercial-pilot instrument requirements
AOPA urges applicants, CFIs: Log time precisely

By Dan Namowitz

On Dec. 17, the FAA issued a formal response to AOPA’s request for clarification on a letter of interpretation (LOI) that stated that hours used to obtain the instrument rating would not count toward the commercial certificate.

“In the response, the FAA confirmed that as long as the training is documented properly, the instrument training received in pursuit of an instrument rating may be counted toward the commercial certificate,” said Kristine Hartzell, AOPA manager of regulatory affairs.

AOPA reported Nov. 4 on Hartzell’s letter to the FAA requesting clarification of its LOI.

AOPA has reviewed the FAA’s clarification and urges instrument pilot applicants and flight instructors to be sure that instrument training is clearly logged to indicate that the training given meets the requirements of 14 CFR 61.65 as well as those of 14 CFR 61.129. That would avoid questions about the training’s applicability should the pilot one day advance to training for a commercial pilot certificate.

AOPA believes that the original FAA interpretation, rendered in response to an inquiry by pilot Richard Theriault, “was not the intent of the requirement of 61.129.” Without this clarification, applicants could have found themselves in the situation of having to acquire additional time to meet the requirements of 14 CFR 61.129 even if they had an instrument rating.

The FAA’s new clarification of that LOI said in part, “We anticipate that for commercial pilot applicants who already hold an instrument rating, the hours of instrument training used to obtain that rating will meet at least some, if not most, or quite often, meet all the requirements for instrument aeronautical experience as required under 61.129.”