Rush orders normally ship the next business day after the order is placed. However, while it's definitely not the norm, some rush items can take longer than expected to be delivered.
Please be aware that items shipped One Day Air and Two Day Air sometimes require up to two days of processing time before they can ship. Rush orders can also only ship and be delivered on non-holiday weekdays within the contiguous United States. And the availability of an item (as described in its listing) is also a factor; some items aren't available for several days.
Again, complications aren't common, but please bear these details in mind when selecting a rush shipping method.