Can you provide a code snippet to avoid confusion?

Would you be able to try the latest POI version - 4.0.0?

Code that reproduce :
File testFile = new File("test.zip");
    OPCPackage mainPkg = OPCPackage.openOrCreate(testFile);

    PackagePartName firstPartName = PackagingURIHelper.createPartName("/dummy/text.txt");
    PackagePart firstPkg = mainPkg.createPart(firstPartName, "text/plain");
    OutputStreamWriter osW = new OutputStreamWriter(firstPkg.getOutputStream(), "UTF8");
    osW.append("FOO");
    osW.flush();
    osW.close();
    firstPkg.flush();
    firstPkg.close();

    PackagePartName secondPartName = PackagingURIHelper.createPartName("/dummy/text.properties");
    PackagePart secondPkg = mainPkg.createPart(secondPartName, "text/plain");
    osW = new OutputStreamWriter(secondPkg.getOutputStream(), "UTF8");
    osW.append("BAR FOO");
    osW.flush();
    osW.close();
    firstPkg.flush();
    firstPkg.close();

    mainPkg.close();

    mainPkg = OPCPackage.openOrCreate(testFile);

And yes it failed also with 4.0.

Exception in thread "main" org.apache.poi.openxml4j.exceptions.InvalidFormatException: The part /dummy/text.properties does not have any content type ! Rule: Package require content types when retrieving a part from a package. [M.1.14]
	at org.apache.poi.openxml4j.opc.ZipPackage$EntryTriple.register(ZipPackage.java:325)
	at org.apache.poi.openxml4j.opc.ZipPackage.getPartsImpl(ZipPackage.java:293)
	at org.apache.poi.openxml4j.opc.OPCPackage.getParts(OPCPackage.java:725)
	at org.apache.poi.openxml4j.opc.OPCPackage.open(OPCPackage.java:238)
	at org.apache.poi.openxml4j.opc.OPCPackage.open(OPCPackage.java:167)
	at org.apache.poi.openxml4j.opc.OPCPackage.openOrCreate(OPCPackage.java:322)
	at test.Main.main(Main.java:59)

I quickly tried to reproduce this, but could not on latest trunk. 

Which operating system are you running this on?

BTW in your sample, you do not flush/close secondPkg, maybe that is related?