We used Jet for a long time and recently switched over to SQLite. Why?
1: When a database gets anywhere near 2 GB or with frequent use, it becomes corrupted in Jet eventually. This has caused us a lot of grief! This has not been fixed in Jet or ACE, though Microsoft has a separate tool that can supposedly fix the database files.
2: Microsoft deprecated Jet years ago, in favor of ACE, but if you read the details, Microsoft itself says that ACE is NOT a replacement for jet, and really wants you to use SQL Server instead.
3: Jet is no longer a standard part of Windows, but part of Microsoft Office, though you can download and install the distributable. However, you can not have both the 32 and 64-bit engines installed at the same time. If you have Office 2007 32-bit installed, and you try and install the 64-bit ACE engine, it tells you need to uninstall Office 2007 first.
So for these reasons we just decided enough is enough. Installing SQL Server is not a solution because it is a big complex invasive install and not very portable.
Our C++ software directly supports SQLite via the sqlite3.c file, and it works very well. I have implemented antive interfaces for OCILIB, Oracle, SQL Server, MySQL etc and this was one of the easiest. It is also much faster than Jet and the resulting files maybe a third of the Jet size. We do have some VB6 and VBA and .NET code that also need to use our database files and for that we use the SQLite ODBC driver (just Google it). Works well.
SQLite works fine in both 32 and 64-bits. And if you read up on it you will see it is seriously tested and amazingly stable. It also supports more standard SQL and is closer to Oracle/SQL Server than Jet is.