FoxPro DBF Index Corruption
There are two basic types of corruption that can occur with CDX indexes: Logical corruption or physical corruption. Based on the OP description that says it is not detected for a while, it sounds like logical corruption.  A common type of logical corruption is when keys either don't  exist for records, or they exist but have incorrect key values.  Physical corruption in CDX indexes typically manifests itself with index pages that are simply invalid (e.g., mismatch between key count and actual keys) or have pointers to other pages that do not exist.  

Physical corruption is usually detected sooner because the application will often produce an error when attempting to process a physically corrupt page.  Logical corruption can go undetected for a longer period of time because it is quite possible to use a logically corrupt index without any errors being reported (e.g., not finding a key in a seek operation is a "normal" situation).

Assuming it is logical corruption being described in the OP, one possible reason for this is if Advantage Database Server is using a different collation than the FoxPro application with which it is sharing the data. The older style CDX indexes do not contain any information in them that defines the collation.  One quick check you might use to see if this is the case is to download the checkindex utility.  It is a simple utility that will scan a table and verify that the keys and records are in agreement with each other.  If you reindex a table with FoxPro, you could then run this utility (which will use Advantage) and determine if they are using the same collation. More information about collations here.