There's no way around it, getting into VFX can be hard. Year on year we see many a fresh face of students applying to join us either at networking events or directly. It can be incredibly disheartening to apply to VFX Studios and not even get so much as an automated email reply. My advice would be threefold:
Firstly, alter your expectations: You may not land the role you had studied or hoped for, you may not intern at that VFX Studio you most admire and you might have to change your expectations and turn on a dime! I joined Framestore as a Runner, with my skills being in Maya as a Generalist; Almost four years later I work as a Houdini Lighting TD and I routinely spend more time excitedly looking at lines of code than rendered pixels on a screen. I didn't know a single line of code when I joined!
Secondly, be prepared to work. Whether it's at completely changing your skillset, being thrown in the deep end or crunching on a project. Having a great work ethic, and a few cups of coffee in you will go a long, long way!
Thirdly, keep working at it - don't give up! There is no such thing as a perfect render, perfect model or perfect animation. You can always improve, learn something new or try and better how you work. Don't be afraid to ask for help or advice, seek out industry professionals and ask their advice. Get feedback on your work and don't take the feedback personally! A standard skill in VFX is being able to take feedback on your work well. Lastly, I know how the silence from the lack of replies from applications can be deafening, but don't let it get to you, if you continue to work at it, you'll be sure to succeed.