Well yes, depending on how its connected & what equipment is used for protection.
Introduction – the technical bit
The most important thing in your bathroom is actually you. Your heart muscles beat around 1 to 2 times a second due to “on-off” electrical impulses sent out by a part of your heart called the “pacemaker”, usually somewhere between 40mV to 70mV.
Your chest muscles (diaphragm) also utilises a similar small mV signal, from your brain.
Your body has a normal “dry” resistance of around 300 Ω, but when it’s wet as in a bathroom it’s lower. The voltage necessary to overcome this resistance is normally around 50V, but when wet can be as low as 25V.
Household electrical systems operate at 230V (a lot larger than the 40 to 70mV mentioned above) with an oscillation of (similar to on-off) 50 times a second.
The idea of designing a safe electrical system is to prevent a voltage across the heart or diaphragm that interrupts these signals, your heart/lungs cannot be allowed beat at 50 times a second, if they did so you would be killed.
Your house electrical system will have a series of devices that trip off the electricity if a fault to earth develops (you may have become part of the fault path to earth).
In older houses this would consist of circuit breakers (fuses or circuit breakers) and earthing conductors, and in your bathroom, there may be “supplementary” earth bonding as well. There are also RCD’s (Residual Current Device’s) which are used in newer electrical installations as well as circuit breakers.
What does this all mean for you – the homeowner
In both cases the time taken to trip is the important thing, not that it does trip. From the information above you could understand that if a fault develops and you are part of the fault path to earth it takes time for your body’s 300 Ω resistance to be overcome, that the 50 oscillation per second wont occur for a few tenths of a second, which is the case.
Hence, its important that the correct things in your house are earthed (including supplementary earth bonding where appropriate) with the correct size of conductor - as this reduces resistance and hence the trip time.