I cannot remember hearing a "dee" sound used like that.
Common diminutives for German names are "-chen" and "-lein". However both border on the ridiculous and are mainly used in jokes or on other special circumstances, much less for forming actual nicknames and such.
So, for example, a Peter might become a Peterchen or a Peterlein. Sometimes you need slight modifications before it works, like inserting a vowel or generating an umlaut: Franz -> Franzilein Franz -> Fränzchen Hans -> Hansilein Hans -> Hänschen