I'm guessing many twitter client that doesn't use browser may not work.
I encountered the same problem when using curl. This problem would happen when web clients use http2 and don't handle Content-Encoding if it don't accept compressed body.
Here is the reproducer.
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$ curl -v --get --http2 --header "Accept-Encoding: identity;q=1.0,*;q=0" https://api.twitter.com/1.1/account/verify_credentials.json
...
GET /1.1/account/verify_credentials.json HTTP/1.1
Host: api.twitter.com
User-Agent: curl/7.50.1
Accept: /
Accept-Encoding: identity;q=1.0,*;q=0
- Connection state changed (MAX_CONCURRENT_STREAMS updated)!
< HTTP/2 400
< content-encoding: gzip
< content-length: 87
< content-type: application/json; charset=utf-8
< date: Tue, 27 Dec 2016 14:05:00 GMT
< server: tsa_m
< set-cookie: guest_id=xxxxxxxxxxxxxxxxxxxxxxxxxxxx
< strict-transport-security: max-age=631138519
< x-connection-hash: xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
< x-response-time: 102
<
- Connection #0 to host api.twitter.com left intact
^_^@^@^@^@^@^@^@\252VJ-\312/V\262\212\256VJ\316OIU\262224\325Q\312M-.NL^G\362\224\234^RS^T^\KK2R\363J2\223^SK2\363\363^TR^RK^R\365\224jck^@^@^@\377\377^C^@^]W^^\342>^@^@^@s$
==============
The response body is encoded with gzip even if I only accept identity. If I use http1.1, this problem doesn't happen.
==============
$ curl -v --get --http1.1 --header "Accept-Encoding: identity;q=1.0,*;q=0" https://api.twitter.com/1.1/account/verify_credentials.json
...
GET /1.1/account/verify_credentials.json HTTP/1.1
Host: api.twitter.com
User-Agent: curl/7.50.1
Accept: /
Accept-Encoding: identity;q=1.0,*;q=0
< HTTP/1.1 400 Bad Request
< content-length: 62
< content-type: application/json; charset=utf-8
< date: Tue, 27 Dec 2016 13:46:59 GMT
< server: tsa_m
< set-cookie: guest_id=xxxxxxxxxxxxxxxxxxxxxxxxx
< strict-transport-security: max-age=631138519
< x-connection-hash: xxxxxxxxxxxxxxxxxxxxxxx
< x-response-time: 101
<
* Connection #0 to host api.twitter.com left intact
{"errors":[{"code":215,"message":"Bad Authentication data."}]}$
===============
================
$ curl --version
curl 7.50.1 (x86_64-pc-linux-gnu) libcurl/7.50.1 GnuTLS/3.5.7 zlib/1.2.8 libidn/1.33 libssh2/1.7.0 nghttp2/1.17.0 librtmp/2.3
Protocols: dict file ftp ftps gopher http https imap imaps ldap ldaps pop3 pop3s rtmp rtsp scp sftp smb smbs smtp smtps telnet tftp
Features: AsynchDNS IDN IPv6 Largefile GSS-API Kerberos SPNEGO NTLM NTLM_WB SSL libz TLS-SRP HTTP2 UnixSockets
================