3:1 Διὸ μηκέτι στέγοντες εὐδοκήσαμεν καταλειφθῆναι ἐν Ἀθήναις μόνοι,
Therefore when we could bear it no longer, we decided to be left alone in Athens;
3:2 καὶ ἐπέμψαμεν Τιμόθεον, τὸν ἀδελφὸν ἡμῶν καὶ συνεργὸν τοῦ θεοῦ ἐν τῷ εὐαγγελίῳτοῦ Χριστοῦ, εἰς τὸ στηρίξαι ὑμᾶς καὶ παρακαλέσαι ὑπὲρ τῆς πίστεως ὑμῶν
and we sent Timothy, our brother and co-worker of God in proclaiming the gospel of Christ, to strengthen and encourage you for the sake of your faith,
3:3 so that no one would be shaken by these persecutions. ...
3:5 διὰ τοῦτο κἀγὼ μηκέτι στέγων ἔπεμψα εἰς τὸ γνῶναι τὴν πίστιν ὑμῶν,
For this reason, when I could bear it no longer I sent to find out about your faith;
17:14 Then the believers immediately sent Paul away to the coast, but Silas and Timothy remained behind.
17:15 οἱ δὲ καθιστάνοντες τὸν Παῦλον ἤγαγον ἕως Ἀθηνῶν, καὶ λαβόντες ἐντολὴν πρὸςτὸν Σιλᾶν καὶ τὸν Τιμόθεον ἵνα ὡς τάχιστα ἔλθωσιν πρὸς αὐτὸν ἐξῄεσαν.
Those who conducted Paul brought him as far as Athens; and after receiving instructions to have Silas and Timothy join him as soon as possible, they left him.
17:16 While Paul was waiting for them in Athens, ...
18:5 When Silas and Timothy arrived from Macedonia, Paul was occupied with proclaiming the word, ...
It is almost invariably assumed that 1 Thess 3:1-2 implies that Paul and Timothy had been in Athens at the same time. This would indeed be in tension with Acts, which says that Timothy stayed behind in Beroea. I propose the following explanation:
Paul and some believers from Beroea travelled to Athens while Timothy and Silas stayed in Beroea. In Athens Paul sent the Beroean believers back to Berea. He was anxious to hear news of the Thessalonian believers so he asked these Beroeans to tell Timothy "come to me soon and visit Thessalonica on the way". Thus, it was these Beroeans who carried the message that sent Timothy to Thessalonica. Their early departure from Athens meant that Paul would be alone there, but it also meant that Paul would soon hear news about the fate of the Thessalonian church. It is this dilemma that Paul describes in 1 Thess 3:3-5. The timing of the return of Paul's Beroean companions to Beroea was determined by Paul's anxiety about Thessalonica.
This explanation creates no conflict between 1 Thess and Acts and has the advantage of parsimony. It adds no events that are in neither text and it adds little to either. Indeed, the only detail missing from Acts is Paul's request to Timothy to make a detour via Thessalonica.
The usual understanding of 1 Thess has Timothy travel to Athens and then make an additional round trip between Athens and Thessalonica. According to ORBIS this round trip by sea was 1080 km, and cost 195 denarii. This is about 6 months wages for a labourer. Paul had to work to earn his keep and this was time that he would surely have rather spent preaching. The missionaries did not have money to spare. It seems unlikely that the team would have chosen to pay for Timothy to make an unnecessary journey back to Thessalonica from Athens. It was more rational for Timothy to stay behind in Macedonia and plan to join Paul latter. The usual interpretation of 1 Thess 3:1-2 has difficulty explaining why Timothy did not visit Thessalonica before leaving Macedonia instead of returning to Thessalonica from Athens.
I anticipate three possible objections:
1. There is nothing in 1 Thess 3:1-2 to suggest that Timothy was not with Paul when Paul sent him to Thessalonica so we can assume that he was. However, this line of reasoning assumes that Paul wrote to record the movements of Timothy. When reading Paul's letters it is easy to forget that they were written for people who knew much more of the background than we do. The audience of 1 Thessalonians already knew the movements of Paul and Timothy. Paul's purpose in writing 1 Thess 3:1-5 was not to record Timothy's journeys but to stress that he (Paul) had been worried about the Thessalonian believers.
2. Acts 17:5 says that Timothy's instructions were to come to Paul ταχέως (as soon as possible). Is this consistent with a detour via Thessalonica? Yes it is. Beroea was only 73 km from Thessalonica, which was the largest city in Macedonia and would be a suitable port from which to find a boat to Athens. Also, the Greek word ταχέως simply means "soon" or "quickly" and does not mean "immediately" in a modern north European sense. In 1 Cor 4:19 Paul says "I will come to you soon (ταχέως)" even though he was going to stay in Ephesus until Pentecost and then make a long detour to Macedonia before travelling to Corinth (1 Cor 16:5-8).
3. Some commentators take the plural verbs in 1 Thess 3:1-2 to refer to Paul and Silas. However, Paul uses the first person singular in 1 Thess 3:5 so, unless he is being inconsistent, he must be using a rhetorical plural in 1 Thess 3:1-2.
I am not the first to suggest that Timothy was never with Paul in Athens. Karl Donfried (Paul, Thessalonica and Early Christianity, 2002, p214) cites Dobschütz  and Rigaux  and writes,
In other words: Paul may have sent Timothy back to Thessalonica either while they were still together in Berea or after Paul arrived in Athens. With regard to the latter, it is possible that those who accompanied Paul from Beroea to Athens (Acts 17.15) presented to Timothy, upon their return, Paul's request. Of these two options, we prefer the former since the latter suggests, dubiously, that Paul had received some new information in Athens.I don't understand why Donfried thinks that Paul would have had to have received new information in Athens. Paul's anxiety about persecutions in Thessalonica would have been heightened by the arrival of Thessalonian opponents in Beroea (Acts 17:13-14). Persecution from these opponents was enough to force Paul's departure to Athens, so Paul would certainly have worried that these same opponents were persecuting the Thessalonian believers. This worry would have continued to trouble Paul in Athens in the absence of new information. 1 Thess 3:1"when we could bear it no longer" implies such a sustained period of anxiety.
Donfried's preference that Paul sent Timothy to Thessalonica from Beroea is unlikely. Paul says that the sending of Timothy resulted in him (Paul) being left alone in Athens. However, the sending of Timothy from Beroea to Thessalonica need not have resulted in Paul being alone in Athens: Paul's Beroean companions or Silas could have stayed with Paul in Athens until Timothy arrived. Furthermore, Paul writes "when I could bear it no longer I sent to find out about your faith" (3:5), but on Donfried's reconstruction he had to bear it for a lot longer anyway. Paul's words make little sense if the interval between the sending of Timothy and Timothy's return to Paul in Corinth with news of Thessalonica (Acts 18:5) was longer than the period of Paul's anxiety about the Thessalonians prior to the sending of Timothy. This makes it unlikely that the sending of Timothy was as early in the sequence as Donfried's preferred option requires.
Others, on the assumption that Timothy was in Athens when Paul sent him, suggest that Acts got it wrong. However, this is special pleading in light of all the details that Acts gets right in this time period (and at other times). Both Acts and 1 Thess have Paul visit Athens after Thessalonica. In both texts Paul has reason to be concerned about persecution in Thessalonica at that time. Both have Paul spending time alone in Athens. Both have Paul eagerly wanting to meet with Timothy. Both have Paul, Silas/Silvanus and Timothy back together later (1 Thess 1:1; Acts 18:5).