I found that one of my ImageButtons stopped sending postback. I've noticed that it uses DoPostBackWithOptions.
I remembered, that I had a problem when my page had PopupControlExtender and Validation controls(see.How do I get rid of DoPostBackWithOptions? ), but it was not a case this time.
I've checked that it is nothing with IIS setting for handling ".axd" extensions - checkbox for "Check that File Exists" is unchecked! (see WebForm_ DoPostBackWithOptions not defined ).
Finally I understood that it is extra html form, that I've copied from prototype. I had the same issue just 2 week ago-Html Form inside ASP.Net page causes no postbacks