Is it possible to compress random sequence with high entropy
Random data can not be compressed. Good pseudo random data can not be compressed(with generic algorithm). 
As Paul commented if you use an inefficient encoding such as hexadecimal or Decimal ASCII characters you are using a full 8 bit character to represent less information. Hexadecimal is compressible to 50% and decimal to $\log_2(10)/8$ which is approx 41%.
If your data is in binary or compresses significantly more than the inherit encoding used.  This is a significant flaw in your random number generator.
A compression test with a Lempel-Ziv based compression algorithm is a good important but not sufficient test. It essentially tests for repeating sequences of arbitrary length. So it can find all sorts of biases in your algorithm.

However for cryptographic security, the chances your homegrown algorithm is secure are slim even it passes all generic statistical tests with flying color.