There are two aspects in the ILR application where bank statements are relevant. Note however that the requirements that apply vary depending on the route you are applying under. The following will be applicable if you are applying as a Tier 1 General Migrant.
1) Evidence for Previous Earnings in the Points Based Assessment - Here the UKBA state that you can claim points for 12 consecutive months of earnings in the 15 month period prior to your application. For this you would need the bank statements for the specific 12 months you are claiming points for.
2) Ability to support yourself & any dependents in your application. - There has been some confusion of what is expected here, and if you search this forum you will find several threads where this question has been raised & answered. The SET(O) form states that you should provide evidence of your finances to prove your ability to maintain and accommodate yourself and any dependants without recourse to public funds. Based on other forum experiences I'd say you don't need your last statement to be within the last calendar month for this. To me, the evidence you provide for previous earnings will satisfy this requirement, as this will clearly indicate that you are earning and therefore can support yourself & any dependents.
See the following for a related application success story where the last bank statement was not provided...
http://www.immigrationboards.com/viewtopic.php?t=101766
Good luck with your application!