With a unknown system, it is possible to measure the sink current for SDA and SCL.
Upload a empty sketch with only Wire.begin() and use a multimeter to measure the shortcut current from SDA to GND and from SCL to GND.
See also my Wiki at Github about the pullup resistors and the sink current.
I'm curious about the gyro working intermittently. What happens when the gyro does not work. Do you mean that the accelerometer always works ? In that case the I2C bus is okay and the fake MPU-6050 has a bad gyro.
Can you show a photo of your project with the wires ?
A breadboard has bad contacts and jumper wires can be broken. So if you use a breadboard, and you look at the wires, assume that every wire and every connection can be bad.