What is the meaning of "400 kHz when other system delays permit?"
I am going through the datasheet of the P82B715 I2C extender.
  The datasheet says "400 kHz when other system delays permit."
  What does this mean? (In the Texas Instruments datasheet they do not mention anything like that.)
  Will it affect my application?


Re "400 kHz when other system delays permit." -> Means: This IC will operate at 400 kHz - in an overall system other components may add delays which drop overall speed to below 400 kHz.
Analogy: A Ferrari in bumper to bumper rush hour traffic is as fast as a Honda Civic and slower than a moped.

The TI datasheet DOES say something like that - but in a different form.

400 kHz = 2.5 uS/clock cycle.
That 1.25 uS = 1250 nS/half cycle.  

trise and tfall for a "barely just making it" pulse is sometimes taken as t_pulse_width/3
here =  about 400 nS.
That is the time allowed for a clocked data level to stabilise.
Longer would be a very good idea.

Note that for this IC the Vin/Vout delay is stated to be 0 ns typical !
This is because there is a 30 Ohm resistor from input to output and the measure of performance is the time for Iout to respond to a change in Vin. 

The TI datasheet section 6.6 page 5 says the output CURRENT step for an input VOLTAGE step - measured from the Vin transitioning Vcc/2 point is 250 nS typical.
A maximum value is not stated BUT 250 ns typ is close enough to 400 ns max to say that the two datasheets are saying much the same thing.


  How can other components can add a delay  


This IC will pass data that arrives properly formed at a 400 kHz rate. Regardless of HOW failure to provide or accept data at this rate is 'not the concern iof the IC designers' - they just tell you what it will do if properly fed.

They say it may be used in star and chain configurations with up to 4000 pF capacitance, and with well under 400 pF per Sx or Sy termination. Your system sounds as if it easily meets these requirements.  

We do not know what is accepting or sourcing data on your network. If they are able to do so at 4 kHz clock rate then it should work at 400 kHz. If not then it won't.



TI P82B715 datasheet
Philips P82B715 datasheet