No ESD diodes?
A likely culprit is RF rectifying by ESD clamp diodes, the charge injection causing latchup, or some kind of malfunction anyway. ("Charge injection" meaning, conduction of said diodes, causing other indiscriminate current flows in the die, in the area around the diode (some ~µm). This can increase supply current, muck up levels of nearby gates, corrupt nearby analog bits including ADC/DAC, PLL (e.g. USB, core clock), etc. 0mA charge injection is recommended on analog pins for example.)
If you don't have a metallic enclosure to ground shields to, and filter everything against, then the next best thing is using the PCB ground plane to the same end. Terminate shields, connectors and filters near the edge, so that the middle can be relatively shielded from outside influences.
Until very high frequencies (GHz), and except for stray resonances (which shouldn't be applicable with most signals and routing), the board is fairly monolithic with respect to outside fields, and problems mostly have to do with wired connections. Conducted emission ends at 30MHz and susceptibility at 80, but it's mostly a practical matter and the same effects continue up into the 100s MHz. For example, your sensor board likely has a resonance in the frequency band of interest, acting as two masses (the boards) joined by a spring (wire harness). Which being unshielded, means about half the radiation goes into signal lines and the other half into ground. Which is why I2C is a terrible idea to go off-board, it has basically no noise immunity, and isn't much better even with filtering.
Shielded cables do nothing unless the shield is grounded. Any voltage induced on the shield, is also induced on the signal lines. If the shield is floating, that voltage just goes straight into whatever's receiving the signals. If the shield is solidly grounded, it shunts that voltage to ground. The important fact is, signals move with respect to the shield; if the shield itself is moving with respect to the circuit, so too will the signals; fix the shield and the signals stay in place. This must be done at both ends. (Occasionally a shield is only grounded at one end, which serves as shielding against electric fields, at low frequencies. It does absolutely nothing for magnetic or radiated induction.)
Tim