It is possible your car never had the full complement of speakers fitted.

In an E39 5-series, there were 6 (radio) and 10 (hifi)-speaker packages.
On the 6-speaker packages you had front door tweeter and bass, but no mid. So there was no speaker fitted behind the mid-range grill. ie: grill was there, but nothing behind it.
In the rear, a full-range speaker was fitted in the parcel tray woofer cavity, and no tweeters in the rear doors.
I thouigh E38 always had all 10 speakers (hifi), with 12 as an option (top-hifi) when the DSP amp was fitted. But maybe you only had 6.

Let's look at the WDS for the wiring for you:

All E38s with non-DSP Amplifier have the following speaker wiring colours:

Front left tweeter: pos =yellow/green, neg = yellow/brown
Front left midrange: pos = yellow/white, neg = yellow/brown
Front left bass: pos = yellow/red, neg = yellow/brown
Front right tweeter: pos =blue/green, neg = blue/brown
Front right midrange: pos = blue/white, neg = blue/brown
Front right bass: pos = blue/red, neg = blue/brown

Rear speakers:
Left tweeter (in door): pos = yellow/black, neg = yellow/green
Left woofer (parcel tray): pos = yellow/violet, neg = yellow/green
Right tweeter (in door): pos = blue/black, neg = blue/green
Right woofer (parcel tray): pos = blue/violet, neg = blue/green

Spot the logic:
Left = yellow, Right = blue
Front: Tweeter = green, mid = white, bass = red
Rear: Tweeter = black, woofer = violet

All speaker wires are twisted into their own pairs.

The wiring from the radio to the amplifier is different for DSP and non-DSP.
On DSP cars, the radio outputs are wired from the front speaker outputs only
This means only 4 wires for sound from radio to DSP amp:
Front left pos, radio pin 1, yellow/red
Front left neg, radio pin 8, brown/orange
Front right pos, radio pin 2, blue/red
Front right neg, radio pin 11, brown/orange

The DSP amp takes care of the fader/balance functions

Without DSP, the radio also has the rear outputs connected (front wiring is same as for DSP):
Rear left pos, radio pin 3, yellow/black
Rear left neg, radio pin 12, brown/orange
Rear right pos, radio pin 6, blue/black
Rear neg, radio pin 14, brown/orange

Without DSP, the amp needs to be switched on by the radio, so check your amp switch-on signal:
Radio pin 16, white. 0V when radio off, +12V when radio on.
With a nav system, the amp is on most of the time.

Check your loom, see if you have the front and rear wires from the radio to the DSP amp. If you have DSP loom and non-DSP amp, you won't get the rear speakers going.

Check your wires and report what you have.
All wire colours are unique, by reporting your wiring colours i will know exactly what they are.

Jochen