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ADTQueue<String> aBucket = new ADTQueue();

Not even sure why it let you get away with the line above as you forget the generic parameter on the right side there too.

Anyway, I think it's...

Throwing numbers into a String array also may be problematic.

If your class is a queue, it should probably be able to hold object itself without the help of an array.

ADTQueue<String>[] buckets = new ADTQueue[10];

Hmmm...that doesn't have the generic on...