Just got caught out by this little mistake for a few minutes. If you call the DbxAccountManager startLink method in your onCreate, onStart or onResume methods for your Activity then the your app will crash after you've authenticated.

Solution: Don't try and link before the activity is fully running. See the Activity API reference for details about the activity lifecycle.

Update: Turns out this has nothing to do with when you start the authentication. After an email exchange with a developer at Dropbox (they contacted me!) we deduced that this was due to a low memory situation, and it just happened by chance that waiting until the activity had fully started fixed the issue. Apparently a fix will be coming for this in a future update...

Here's the error from the logcat:

E/AndroidRuntime(23572): java.lang.RuntimeException: Unable to resume activity {your.namespace/com.dropbox.sync.android.DbxAuthActivity}: java.lang.IllegalStateException: Dropbox isn't initialized.
E/AndroidRuntime(23572):        at android.app.ActivityThread.performResumeActivity(ActivityThread.java:2567)
E/AndroidRuntime(23572):        at android.app.ActivityThread.handleResumeActivity(ActivityThread.java:2595)
E/AndroidRuntime(23572):        at android.app.ActivityThread.handleLaunchActivity(ActivityThread.java:2109)
E/AndroidRuntime(23572):        at android.app.ActivityThread.access$600(ActivityThread.java:132)
E/AndroidRuntime(23572):        at android.app.ActivityThread$H.handleMessage(ActivityThread.java:1157)
E/AndroidRuntime(23572):        at android.os.Handler.dispatchMessage(Handler.java:99)
E/AndroidRuntime(23572):        at android.os.Looper.loop(Looper.java:137)
E/AndroidRuntime(23572):        at android.app.ActivityThread.main(ActivityThread.java:4575)
E/AndroidRuntime(23572):        at java.lang.reflect.Method.invokeNative(Native Method)
E/AndroidRuntime(23572):        at java.lang.reflect.Method.invoke(Method.java:511)
E/AndroidRuntime(23572):        at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:789)
E/AndroidRuntime(23572):        at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:556)
E/AndroidRuntime(23572):        at dalvik.system.NativeStart.main(Native Method)
E/AndroidRuntime(23572): Caused by: java.lang.IllegalStateException: Dropbox isn't initialized.
E/AndroidRuntime(23572):        at com.dropbox.sync.android.DbxAccountManager.getInstance(DbxAccountManager.java:144)
E/AndroidRuntime(23572):        at com.dropbox.sync.android.DbxAuthActivity.finishAuth(DbxAuthActivity.java:72)
E/AndroidRuntime(23572):        at com.dropbox.sync.android.DbxAuthActivity.onResume(DbxAuthActivity.java:51)
E/AndroidRuntime(23572):        at android.app.Instrumentation.callActivityOnResume(Instrumentation.java:1154)
E/AndroidRuntime(23572):        at android.app.Activity.performResume(Activity.java:4539)
E/AndroidRuntime(23572):        at android.app.ActivityThread.performResumeActivity(ActivityThread.java:2557)
E/AndroidRuntime(23572):        ... 12 more