Depends on the registration date and the date in February. Using timeanddate.com 6 months earlier than  February 28, 2010  is August 28, 2009. (182.5 days or 26 weeks is August 30th).   Using the  technical   date calculation, It is possible that school started and a boy joined  on or before August 28.

Your mileage may vary on the rest of this.  My experiences with my council on cub scouts is  if the scout  registered in August all of August counts, etc.  So  using this looser method  August  to February  is six months.  I have heard from  cub-scout-talk that national considers a paid registration to be an active cub scout.  

We have a boy in our pack this year that is working on this very thing.

 Meritbadge.org has a whole page dedicated to the confusion of the definition of Active in the BSA (http://meritbadge.org/wiki/ind…..php/Active) Meritbadge.org actually comments that the Definition for Active under the Webelos badge is more restrictive than  the Boy Scout definition.

Scibd.com has a document that  is reportedly  item #57-492 from the BSA. "Rules and Regulations of the Boy Scout of America".  On image 7  under Article VII. Youth Members, General Classifications, Section 1, Active, Clause 1; there is yet another definition of the word Active.

So the  scouts do not have a consistent definition for active. Do you hold the scout  to the  more resctrictive  one for the Webelos badge or  any of the other less restrictive definitions.  When determining "Active", my answer is  you're not wrong any where from "a registered scout is an active scout" and "having good attendance, paying your den dues, working on den projects."  There is precedent for both. I have the feeling that if the scout was registered in  September and the crossover is in late February, the council won't give you any static. As you pointed out the professional scouters often say "it's not like its Eagle required."

I do not think the time requirement for Webelos and Arrow of light  are consecutive months.  From reading the other requirements, specifically the troop outdoor activity, it states the same  activity cannot be used for Outdoorsman and Arrow of Light.  If consecutive  I believe they would state it as such. Consecutive  time in rank  would also change the Arrow of Light requirements for units that base rank strictly on age such as the LDS or possibly a home school only  unit.  

In these LDS or age based units the boys  move up in rank throughout the year as their birthday occurs.  So  when a boy turns 10 he is a webelos. Meeting the time requirements for the Webelos Badge  and the Arrow of light concurrently concurrently would allow the scout to earn his arrow of light at 10 years and 6 months, as stated in the Arrow of Light requirement.  Having to meet the time in den  requirements consecutively, would mean  the earliest the  age based scouting unit could award the  Arrow of light is when the boy is 10 years and 9 months.

On the School based side consecutive time in den requirements  would mean most  boys joining in the 5th grade would never earn the AOL  or earn it in May.

So  to my logic the Webelos and Arrow of Light  time in den requirement is  or can be running concurrently provided the  scout is in the fifth grade or 10 years old  (*and registered in an age based unit).  Still for many this puts the earliest to  receive the AOL and join in September of the fifth grade as March 1

* This  proviso for being registered in an age based unit  is based on my original question and Shawn's answer that no 4th grade  cub scout could earn the Arrow of Light.

I hope you've been able to follow my logic and understand how I came to my answers.