Why should you definitely not skip Negative testing ?

What is Negative Testing ?

Negative Testing:

As software testing professionals we often counter situations where we need to test an application negatively. Here we check whether the system behaves properly (as expected) even when we provide invalid data to the system.
This tests the robustness of system against all odds (negative scenarios).

A user might enter invalid data by mistake , but the software should be well equipped to handle this. 

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Negative testing scenarios

You want to login to your email account.
And you possibly forgot your password. The moment you enter a password(most likely incorrect one), you will get a notice from the application that either of your username and password is incorrect.
Now imagine, if you were not getting this notice from the application and you go on entering wrong login details in to the system over and over. It would be quite frustrating to not get access to your account and even without a clue as to what went wrong.
Wouldn't you like the application to be more interactive so that you know what happening at the back end ?

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Negative test

Testing the application with an intent to uncover such defects which may be the outcome of an invalid input.
Then tuning the system to display proper error messages. Also to make sure that system does not behave totally awkward when it encounters such invalid inputs. Such as there should be not crash of application in any case etc.