Four cities is a bit too much for just <10 days.... at least if you want to explore the surroundings too, which is highly recommended since all the cities you mentioned, except Leipzig, were completely leveled in WW2, and the smaller towns nearby usually were not. And Leipzig was mostly rebuilt in the 19th century.
Here is a map of Germanys highlights according to Baedeker, Germanys most popular travel guide:
http://i.imgur.com/9BcKenL.jpg
As you can see, all the mentioned cities offer plenty of sights nearby. In case of Dresden preserved towns like Bautzen, Görlitz, Meissen, Freiberg or Pirna, moreover castles, fortresses and palaces, and of course Saxon Switzerland. From Leipzig you could see Wittenberg (Luther), Naumburg, Weimar (Goethe and Schiller), the Dessau-Wörlitz Garden Realm or the towns of the Harz mountains. And Hamburg is great in combination with the Baltic coast, Germanys most popular summer destination, which offers beautiful towns and resorts, and great beaches.
I'm not a big fan of Nuremberg, which to me is just another poorly rebuilt post-war city with some great sights like the castle area, the churches and two or three museums. But it's a good base for day trips to towns nearby, like Rothenburg, Bamberg or Regensburg.
I would recommend to drop 2 of the 4 cities in favor of some day trips. Either see
- Hamburg and the Baltic coast
- Dresden and Leipzig
- Leipzig or Dresden and Nuremberg
- Hamburg or Leipzig or Dresden and the Harz mountains
The Harz mountains are my favorite part of Germany. They offer many perfectly preserved towns full half-timbered houses. Two of them, Goslar and Quedlinburg, are World Heritage Sites. Quedlinburg is like a bigger and less touristy Rothenburg. Here you can read what others have to say about the town:
http://www.tripadvisor.com/AttractionReview-g198402-d2239030-Reviews-AltstadtQuedlinburg-QuedlinburgSaxonyAnhalt.html#REVIEWS
And here is a gallery with 130.000 pics of the Harz mountains:
http://www.raymond-faure.com/indexharzenglish.htm
(have a look at Quedlinburg, Goslar, Wernigerode, Stolberg, Wolfenbüttel, Celle or the cathedral of Halberstadt)