Accrued interest is interest that has accrued on the Fractional Bond but has not yet been paid. 

Interest accrues from the date the last interest payment was made until it is paid at the next interest payment date.

When you purchase a Fractional Bond you pay the seller the accrued interest up to the date of the purchase. Since you will receive the full coupon at the next interest payment date, you will have earned interest based on how long you have held the Fractional Bond.

Interest continues to accrue on Fractional Bonds even if they are in default but ultimately whether or not this is paid is dependent on whether the underlying bond eventually becomes impaired.