Update
Another issue we ran into was related to client machines having an outdated cert. Once the updated cert was published to SharePoint, the client machines downloaded the new cert and were able to upload to SharePoint.

Ran into a small issue when testing some code for a SharePoint 2010 to SharePoint 2016 migration.

With SharePoint 2010, the following code sample would work to upload files to a library.

//ServRef is a Service Reference to _vti_bin/ListData.svc
string sharePointSvc = "https://sp2016.some.url/sites/random/_vti_bin/ListData.svc";

            using (FileStream file = File.Open(@"C:\test1.docx", FileMode.Open))
            {
                ServRef.RandomDataContext ctx = new ServRef.RandomDataContext(new Uri(sharePointSvc));

                ctx.Credentials = System.Net.CredentialCache.DefaultNetworkCredentials;

                string filename = Path.GetFileNameWithoutExtension(file.Name);
                string path = "/sites/random/dropofflibrary/" + Path.GetFileName(file.Name);
                string contentType = "Interest Summary";

                ServRef.DropOffLibraryItem documentItem = new ServRef.DropOffLibraryItem()
                {
                    ContentType = contentType,
                    Name = filename,
                    Title = filename
                };

                ctx.AddToDropOffLibrary(documentItem);
                ctx.SetSaveStream(documentItem, file, false, contentType, path);

                try
                {
                    ctx.SaveChanges();
                }
                catch (Exception ex)
                {
                    var err = ex.Message;
                    throw;
                }
            } 

When trying to use this same code with SharePoint 2016, I was receiving the following Errors:
Output error: An error occurred while processing this request.

Errors from Fiddler:
Auth:
No Proxy-Authenticate Header is present.

No WWW-Authenticate Header is present.

Caching:

Under RFC2616, HTTP/500 responses will not be cached regardless of what caching headers may be present. HTTP/1.1 Cache-Control Header is present: no-cache

This response does not specify explicit HTTP Cache Lifetime information and does not specify a Last-Modified date. Heuristic expiration is typically based on Last-Modified date. Lacking Last-Modified, this response may be revalidated on every use or once per browsing session, depending on the browser configuration.

This response contains neither an ETAG nor a Last-Modified time. This will prevent a Conditional Revalidation of this response.

FIX
For SharePoint 2016, the upload process required that the Path property be populated.

//ServRef is a Service Reference to _vti_bin/ListData.svc
string sharePointSvc = "https://sp2016.some.url/sites/random/_vti_bin/ListData.svc";

            using (FileStream file = File.Open(@"C:\test1.docx", FileMode.Open))
            {
                ServRef.RandomDataContext ctx = new ServRef.RandomDataContext(new Uri(sharePointSvc));

                ctx.Credentials = System.Net.CredentialCache.DefaultNetworkCredentials;

                string filename = Path.GetFileNameWithoutExtension(file.Name);
                string path = "/sites/random/dropofflibrary/" + Path.GetFileName(file.Name);
                string contentType = "Interest Summary";

                ServRef.DropOffLibraryItem documentItem = new ServRef.DropOffLibraryItem()
                {
                    Path = path,
                    ContentType = contentType,
                    Name = filename,
                    Title = filename
                };

                ctx.AddToDropOffLibrary(documentItem);
                ctx.SetSaveStream(documentItem, file, false, contentType, path);

                try
                {
                    ctx.SaveChanges();
                }
                catch (Exception ex)
                {
                    var err = ex.Message;
                    throw;
                }
            }