Nginx blocks file upload to static page: 405 Not Allowed - file-upload

Question

I'm having a problem with my Nginx server. I installed a new server and thought I put up the same installation as on my old one. However, since then, I cannot upload files to static pages. I had an upload function in an index.php. It still works when I explicitly call index.php, but when I just call /, it gives me a 405 error.
I have seen numerous discussions on the internet where it all comes down to nginx not allowing POST requests to static files. However, my installation accepts POST requests to static files, but no file uploads. To clarify, this is my current situation:
+----------------------+-------------+------------------+
| | Common POST | With file upload |
+----------------------+-------------+------------------+
| Request to / | Allowed | Allowed |
| Request to index.php | Allowed | 405 Not Allowed |
+----------------------+-------------+------------------+
It probably doesn't add anything, but for the sake of completeness: I'm running a WordPress site on this server as well, and since migrating the server I have not been able to upload new media.
The logfiles (/var/log/nginx/error.log, /var/log/php5-fpm.log, /var/log/syslog) don't give any information. Later, I enabled logging for the FastCGI pool as well, and there in the access log, there is no record of the POST requests to the static page. So nginx must block them before they get to FastCGI.
The relevant PHP ini settings:
enable_post_data_reading On
file_uploads On
max_file_uploads 20
upload_max_filesize 20M
upload_tmp_dir no value
How can I make it work again?
Additional information
There are some differences between my old server and the new server (maybe more, but I'm not aware):
In nginx, before, I had to use fastcgi_pass 127.0.0.1:9000; - now, this doesn't work anymore, but I'm using fastcgi_pass unix:/var/run/php5-fpm.sock;. I'm not sure why this is or if it matters
I built nginx from source. Here are the different installations, first the old one:
$ nginx -V
nginx version: nginx/1.4.6 (Ubuntu)
built by gcc 4.8.2 (Ubuntu 4.8.2-16ubuntu6)
TLS SNI support enabled
configure arguments: --with-cc-opt='-g -O2 -fstack-protector --param=ssp-buffer-size=4 -Wformat -Werror=format-security -D_FORTIFY_SOURCE=2' --with-ld-opt='-Wl,-Bsymbolic-functions -Wl,-z,relro' --prefix=/usr/share/nginx --conf-path=/etc/nginx/nginx.conf --http-log-path=/var/log/nginx/access.log --error-log-path=/var/log/nginx/error.log --lock-path=/var/lock/nginx.lock --pid-path=/run/nginx.pid --http-client-body-temp-path=/var/lib/nginx/body --http-fastcgi-temp-path=/var/lib/nginx/fastcgi --http-proxy-temp-path=/var/lib/nginx/proxy --http-scgi-temp-path=/var/lib/nginx/scgi --http-uwsgi-temp-path=/var/lib/nginx/uwsgi --with-debug --with-pcre-jit --with-ipv6 --with-http_ssl_module --with-http_stub_status_module --with-http_realip_module --with-http_addition_module --with-http_dav_module --with-http_geoip_module --with-http_gzip_static_module --with-http_image_filter_module --with-http_spdy_module --with-http_sub_module --with-http_xslt_module --with-mail --with-mail_ssl_module --add-module=/build/buildd/nginx-1.4.6/debian/modules/nginx-auth-pam --add-module=/build/buildd/nginx-1.4.6/debian/modules/nginx-dav-ext-module --add-module=/build/buildd/nginx-1.4.6/debian/modules/nginx-echo --add-module=/build/buildd/nginx-1.4.6/debian/modules/nginx-upstream-fair --add-module=/build/buildd/nginx-1.4.6/debian/modules/ngx_http_substitutions_filter_module
And the new one:
$ nginx -V
nginx version: nginx/1.6.0
built by gcc 4.8.2 (Ubuntu 4.8.2-19ubuntu1)
TLS SNI support enabled
configure arguments: --with-cc-opt='-g -O2 -fstack-protector --param=ssp-buffer-size=4 -Wformat -Werror=format-security -D_FORTIFY_SOURCE=2' --with-ld-opt='-Wl,-Bsymbolic-functions -Wl,-z,relro' --conf-path=/etc/nginx/nginx.conf --http-log-path=/var/log/nginx/access.log --error-log-path=/var/log/nginx/error.log --lock-path=/var/lock/nginx.lock --pid-path=/run/nginx.pid --http-client-body-temp-path=/var/lib/nginx/body --http-fastcgi-temp-path=/var/lib/nginx/fastcgi --http-proxy-temp-path=/var/lib/nginx/proxy --http-scgi-temp-path=/var/lib/nginx/scgi --http-uwsgi-temp-path=/var/lib/nginx/uwsgi --with-debug --with-pcre-jit --with-ipv6 --with-http_ssl_module --with-http_stub_status_module --with-http_realip_module --with-http_addition_module --with-http_dav_module --with-http_geoip_module --with-http_gzip_static_module --with-http_image_filter_module --with-http_spdy_module --with-http_sub_module --with-http_xslt_module --with-mail --with-mail_ssl_module --add-module=/home/camilstaps/nginx-modules/nginx-auth-pam --add-module=/home/camilstaps/nginx-modules/nginx-dav-ext-module --add-module=/home/camilstaps/nginx-modules/ngx_http_substitutions_filter_module --with-http_mp4_module --with-http_flv_module
The old PHP version:
$ php5 -v
PHP 5.5.9-1ubuntu4.3 (cli) (built: Jul 7 2014 16:36:58)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.5.0, Copyright (c) 1998-2014 Zend Technologies
with Zend OPcache v7.0.3, Copyright (c) 1999-2014, by Zend Technologies
And the new one:
$ php5 -v
PHP 5.5.9-1ubuntu4.3 (cli) (built: Jul 7 2014 16:36:58)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.5.0, Copyright (c) 1998-2014 Zend Technologies
Lastly, I still have access to my old installation. So if you'd like to see the content of some file or the output of some command that's possible.

Answer 1

The problem turned out to be that I had put directives from the http_mp4_module in the location / block:
location / {
...
mp4;
}
Apparently, this gives a conflict somehow.
I removed the directives from this block and put them in a location ~ \.mp4 block.