Your problem is undoubtedly a matter of signal to noise ratio.

LSR meters operate as Timfox explained in post #13. You can also find a good in depth explanation in the Agilent Impedance measurement handbook. Google "impedance measurement handbook"

When you measure a capacitor, the meter applies a sine wave of whatever the nominal voltage is (1 volt for your measurement) and extracts the in phase and quadrature components of the resulting current. The ratio of the two components is equal to the value of D.

So if your capacitor is low loss, the real part of the current is going to be very small. For example, if the cap has a D of .001, the meter will be trying to measure a very small current/voltage (microvolts). When the measured value of D < .001 (or equivalently, Q>1000), low cost meters are going to have problems.

A good meter will know when it's measurement is in the noise, and will not show very many digits for the result. Watch out for that clue that the measurement is "iffy". For example in reply #9 your measurement shows 4 digits for the capacitance, but only 2 digits for the ESR.

In reply #15, the 10 kHz measurement of ESR shows a value of .0085 ohms. This is impossible. The value of Q is Xc/ESR; that measurement would imply a Q of about 188000.

The Hioki IM3570 can be set to show 7 digits. This is well beyond the accuracy of the instrument but it's handy for matching components, and watching drift with temperature. Here's what I get when I measure a 10 nF silver mica capacitor, which is a very low loss capacitor, at 10 kHz:

The capacitance is shown as 7 digits, but only about 5 of them are stable. ESR (Rs) is shown with 6 digits, but only the leading 2 digits are stable. Note that the Q is about 6000; this is approaching the capability of the instrument. When the Q>10000, the Hioki won't even show a value; it just indicates an out of range value.

I think that your East Tester just can't cope with very low loss capacitors. I've noticed that some of the hand held meters I have exhibit the same limitations.