After I tried everything noted here without success I stumbled over some other information and for now, it works.
I'd like to share the information (note: I tried PRAM etc and stopping processes that communicate to the outside etc)
What finally did it was this
1
Checking the sleep states with the Terminal command
pmset -g
2
Setting the hibernatemode to 3 (corresponds to suspend to RAM and to disk) with the Terminal command
sudo pmset hibernatemode 3
->you may be prompted for your password.
If you run your Mac without one, it will keep prompting you, so you have to change the password for the time being (System settings -> users & Groups [I translate that from german so this might sound slightly different on your system])
(hibernatemode 3 seems to be the default value on new Macs. Mine is some 2 years old and was on 0)
3
check setting with
pmset -g
At this point, my Mac went in screen sleep mode, but the HD kept spinning, so I did the following
4
pmset -a ttyskeepawake 0
5
check again with
pmset -g
Which displayed this:
Active Profiles:
Battery Power -1
AC Power -1*
Currently in use:
womp 0
halfdim 0
sms 1
panicrestart 157680000
gpuswitch 2
hibernatefile /var/vm/sleepimage
networkoversleep 0
disksleep 0
sleep 180
hibernatemode 3
ttyskeepawake 0
displaysleep 180
acwake 0
lidwake 1
6
Close the lid or use the sleep command under the apple menu
7
Slowly count to ten
(It takes that long on my MacBook to fall really asleep)
References:
Set newer portable Macs' sleep mode | Macworld
DropBox for Mac is preventing my MacBook Pro from sleeping « Dropbox Forums post by user Tom H.
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I hope this is of use for some other Mac user.