When will a TCP network packet be fragmented at the application layer?

It will be split when it hits a network device with a lower MTU than the packet's size. Most ethernet devices are 1500, but it can often be smaller, like 1492 if that ethernet is going over PPPoE (DSL) because of the extra routing information, even lower if a second layer is added like Windows Internet Connection Sharing. And dialup is normally 576!

In general though you should remember that TCP is not a packet protocol. It uses packets at the lowest level to transmit over IP, but as far as the interface for any TCP stack is concerned, it is a stream protocol and has no requirement to provide you with a 1:1 relationship to the physical packets sent or received (for example most stacks will hold messages until a certain period of time has expired, or there are enough messages to maximize the size of the IP packet for the given MTU)

As an example if you sent two "packets" (call your send function twice), the receiving program might only receive 1 "packet" (the receiving TCP stack might combine them together). If you are implimenting a message type protocol over TCP, you should include a header at the beginning of each message (or some other header/footer mechansim) so that the receiving side can split the TCP stream back into individual messages, either when a message is received in two parts, or when several messages are received as a chunk.

The question makes an assumption that is not true -- TCP does not deliver packets to its endpoints, rather, it sends a stream of bytes (octets). If an application writes two strings into TCP, it may be delivered as one string on the other end; likewise, one string may be delivered as two (or more) strings on the other end.

RFC 793, Section 1.5:

"The TCP is able to transfer a continuous stream of octets in each direction between its users by packaging some number of octets into segments for transmission through the internet system."

The key words being continuous stream of octets (bytes).

RFC 793, Section 2.8:

"There is no necessary relationship between push functions and segment boundaries. The data in any particular segment may be the result of a single SEND call, in whole or part, or of multiple SEND calls."

The entirety of section 2.8 is relevant.

Fragmentation should be transparent to a TCP application. Keep in mind that TCP is a stream protocol: you get a stream of data, not packets! If you are building your application based on the idea of complete data packets then you will have problems unless you add an abstraction layer to assemble whole packets from the stream and then pass the packets up to the application.

At the application layer there are any number of reasons why the whole 1500 bytes may not show up one read. Various factors in the internal operating system and TCP stack may cause the application to get some bytes in one read call, and some in the next. Yes, the TCP stack has to re-assemble the packet before sending it up, but that doesn't mean your app is going to get it all in one shot (it is LIKELY will get it in one read, but it's not GUARANTEED to get it in one read).

TCP tries to guarantee in-order delivery of bytes, with error checking, automatic re-sends, etc happening behind your back. Think of it as a pipe at the app layer and don't get too bogged down in how the stack actually sends it over the network.