Could I Connect many Arduino Nano boards together?
I want to know if I could connect many Arduino Nano boards together...


Yes, you can read about it here at arduino.cc. Where is says: "...it can be helpful to set up two (or more!) Arduino ... boards to share information with each other..."


  ... i want to know if there is any problem in Communication between
  them using I2C ...


This feature, according to the above link, is expected to work.  (Can you explain the type of problems you are avoiding?)


  ...could [i2c communication] affect processing speed on every board ...


Most likely it will.  Most Arduino platforms do not run operating systems.  So your program is all that is running.  If your program is busy sampling sensors, it will be difficult (not impossible) to concurrently handle i2c communications.